A comparison between nonparametric estimators for finite population distribution functions 5. Simulation study

In this section we analyze some simulation results. Our goal is to compare efficiency with respect to the sample design of the distribution function estimators introduced in Section 2 and of the variance estimators of Section 4. The simulation results refer to simple random without replacement sampling and to Poisson sampling with unequal inclusion probabilities. As a benchmark, we included also the Horvitz-Thompson distribution function estimator

F ^ π ( t ) : = 1 N j s π j 1 I ( y j t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaja WaaSbaaSqaaiabec8aWbqabaGcdaqadaqaaiaadshaaiaawIcacaGL PaaacaaI6aGaaGypamaalaaabaGaaGymaaqaaiaad6eaaaWaaabuae aacqaHapaCdaqhaaWcbaGaamOAaaqaaiabgkHiTiaaigdaaaGccaWG jbWaaeWaaeaacaWG5bWaaSbaaSqaaiaadQgaaeqaaOGaeyizImQaam iDaaGaayjkaiaawMcaaaWcbaGaamOAaiabgIGiolaadohaaeqaniab ggHiLdaaaa@4EB8@

and the corresponding variance estimator

V ˜ ( F ^ π ( t ) ) := 1 N 2 i , j s π i , j π i π j π i , j π i π j I ( y i t ) I ( y j t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOvayaaia WaaeWaaeaaceWGgbGbaKaadaWgaaWcbaGaeqiWdahabeaakmaabmaa baGaamiDaaGaayjkaiaawMcaaaGaayjkaiaawMcaaiaaiQdacaaI9a WaaSaaaeaacaaIXaaabaGaamOtamaaCaaaleqabaGaaGOmaaaaaaGc daaeqbqabSqaaiaadMgacaaISaGaamOAaiabgIGiolaadohaaeqani abggHiLdGcdaWcaaqaaiabec8aWnaaBaaaleaacaWGPbGaaGilaiaa dQgaaeqaaOGaeyOeI0IaeqiWda3aaSbaaSqaaiaadMgaaeqaaOGaeq iWda3aaSbaaSqaaiaadQgaaeqaaaGcbaGaeqiWda3aaSbaaSqaaiaa dMgacaaISaGaamOAaaqabaGccqaHapaCdaWgaaWcbaGaamyAaaqaba GccqaHapaCdaWgaaWcbaGaamOAaaqabaaaaOGaamysamaabmaabaGa amyEamaaBaaaleaacaWGPbaabeaakiabgsMiJkaadshaaiaawIcaca GLPaaacaWGjbWaaeWaaeaacaWG5bWaaSbaaSqaaiaadQgaaeqaaOGa eyizImQaamiDaaGaayjkaiaawMcaaaaa@6BF8@

in the simulation study.

We considered both artificial and real populations. The former were obtained by generating N = 1,000 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOtaiaai2 dacaqGXaGaaeilaiaabcdacaqGWaGaaeimaaaa@3A15@ values x i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBa aaleaacaWGPbaabeaaaaa@3716@ from i.i.d. uniform random variables with support on the interval ( 0,1 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaaca aIWaGaaGilaiaaigdaaiaawIcacaGLPaaaaaa@38B3@ and by combining them with three types of regression function m ( x ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBamaabm aabaGaamiEaaGaayjkaiaawMcaaaaa@3877@ and two types of error components ε i . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyTdu2aaS baaSqaaiaadMgaaeqaaOGaaiOlaaaa@387C@ The regression functions are (i) m ( x ) = 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBamaabm aabaGaamiEaaGaayjkaiaawMcaaiaai2dacaaIWaaaaa@39F8@ (flat), (ii) m ( x ) = 10 x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBamaabm aabaGaamiEaaGaayjkaiaawMcaaiaai2dacaaIXaGaaGimaiaadIha aaa@3BB0@ (linear) and (iii) m ( x ) = 10 x 1 / 4 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBamaabm aabaGaamiEaaGaayjkaiaawMcaaiaai2dacaaIXaGaaGimaiaadIha daahaaWcbeqaamaalyaabaGaaGymaaqaaiaaisdaaaaaaaaa@3D6C@ (concave), while the error components ε i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyTdu2aaS baaSqaaiaadMgaaeqaaaaa@37C0@ are either independent realizations from a unique Student t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiDaaaa@35F8@ distribution with ν = 5 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyVd4MaaG ypaiaaiwdaaaa@383D@ d.o.f., or independent realizations from N MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOtaaaa@35D2@ different shifted noncentral Student t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiDaaaa@35F8@ distributions with ν = 5 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyVd4MaaG ypaiaaiwdaaaa@383D@ d.o.f. and with noncentrality parameters given by μ = 15 x i . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiVd0MaaG ypaiaaigdacaaI1aGaamiEamaaBaaaleaacaWGPbaabeaakiaac6ca aaa@3BC9@ The shifts applied to the error components in the latter case make sure that the means of the noncentral Student t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiDaaaa@35F8@ distributions from which they were generated are zero. The artificial populations are shown in Figure 5.1 to 5.3. As for the real populations, we took the M U 2 8 4 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamytaiaadw faieGacaWFYaGaa8hoaiaa=rdaaaa@38D7@ Population of Sweden Municipalities of Särndal et al. (1992) (population size N = 284 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeGaaeaaca WGobGaaGypaiaaikdacaaI4aGaaGinaaGaayzkaaaaaa@399D@ and considered the natural logarithm of R M T 8 5 = MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOuaiaad2 eacaWGubacbiGaa8hoaiaa=vdacqGH9aqpaaa@3A01@ Revenues from the 1985 municipal taxation (in millions of kronor) as study variable Y , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamywaiaacY caaaa@368D@ and the natural logarithm of either P 8 5 = 1 9 8 5 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiuaGqaci aa=HdacaWF1aGaeyypa0Jaa8xmaiaa=LdacaWF4aGaa8xnaaaa@3B2F@ population (in thousands) or R E V 8 4 = MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOuaiaadw eacaWGwbacbiGaa8hoaiaa=rdacqGH9aqpaaa@39FA@ Real estate values according to 1984 assessment (in millions of kronor) as auxiliary variable X . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiwaiaac6 caaaa@368E@ The real populations are shown in Figure 5.4.

Figure 5.1 of article 14541

Description of Figure 5.1

Figure made of two scatter plots ( y MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaqabaqaai aadMhaaiaawIcaaaaa@3813@  versus x ), MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaqacaqaai aadIhaaiaawMcaaiaacYcaaaa@38C4@  each one illustrating an artificial population. The first graph is the population generated from y i = ε i , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWG5bWaaS baaSqaaiaadMgaaeqaaOGaeyypa0JaeqyTdu2aaSbaaSqaaiaadMga aeqaaOGaaiilaaaa@3CF2@  where ε i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH1oqzda WgaaWcbaGaamyAaaqabaGccqWI8iIoaaa@3A43@  i.i.d. Student t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWG0baaaa@3748@  with ν=5. MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH9oGBcq GH9aqpcaaI1aGaaiOlaaaa@3A7E@  The y-axis goes from -4 to 8 and the x-axis goes from 0.0 to 1.0. The scatter plot is centered around y=0. MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpepeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuj0lXxdrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWG5bGaey ypa0JaaGimaiaac6caaaa@3BBF@  The second graph is the population generated from y i = ε i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWG5bWaaS baaSqaaiaadMgaaeqaaOGaeyypa0JaeqyTdu2aaSbaaSqaaiaadMga aeqaaaaa@3C38@  and ε i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH1oqzda WgaaWcbaGaamyAaaqabaGccqWI8iIoaaa@3A43@  indep. noncentral Student t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWG0baaaa@3748@  with ν=5 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH9oGBcq GH9aqpcaaI1aaaaa@39CC@  and μ=15 x i . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH8oqBcq GH9aqpcaaIXaGaaGynaiaadIhadaWgaaWcbaGaamyAaaqabaGccaGG Uaaaaa@3D58@  The y-axis goes from -10 to 40 and the x-axis goes from 0.0 to 1.0. The scatter plot is concentrated around y=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpepeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuj0lXxdrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWG5bGaey ypa0JaaGimaaaa@3B0D@  for small values of x. MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpepeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuj0lXxdrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWG4bGaai Olaaaa@39FE@  The variation increases when x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpepeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuj0lXxdrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWG4baaaa@394C@  increases.

Figure 5.2 of article 14541

Description of Figure 5.2

Figure made of two scatter plots ( y MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaqabaqaai aadMhaaiaawIcaaaaa@3813@  versus x ), MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaqacaqaai aadIhaaiaawMcaaiaacYcaaaa@38C4@  each one illustrating an artificial population. The first graph is the population generated from y i =10 x i + ε i , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWG5bWaaS baaSqaaiaadMgaaeqaaOGaeyypa0JaaGymaiaaicdacaWG4bWaaSba aSqaaiaadMgaaeqaaOGaey4kaSIaeqyTdu2aaSbaaSqaaiaadMgaae qaaOGaaiilaaaa@416A@  where ε i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH1oqzda WgaaWcbaGaamyAaaqabaGccqWI8iIoaaa@3A43@  i.i.d. Student t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWG0baaaa@3748@  with ν=5. MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH9oGBcq GH9aqpcaaI1aGaaiOlaaaa@3A7E@  The y-axis goes from 0 to 10 and the x-axis goes from 0.0 to 1.0. The scatter plot is showing an increasing linear relationship between x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpepeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuj0lXxdrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWG4baaaa@394C@  and y. MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpepeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuj0lXxdrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWG5bGaai Olaaaa@39FF@  The second graph is the population generated from y i = ε i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWG5bWaaS baaSqaaiaadMgaaeqaaOGaeyypa0JaeqyTdu2aaSbaaSqaaiaadMga aeqaaaaa@3C38@  and ε i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH1oqzda WgaaWcbaGaamyAaaqabaGccqWI8iIoaaa@3A43@  indep. noncentral Student t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWG0baaaa@3748@  with ν=5 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH9oGBcq GH9aqpcaaI1aaaaa@39CC@  and μ=15 x i . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH8oqBcq GH9aqpcaaIXaGaaGynaiaadIhadaWgaaWcbaGaamyAaaqabaGccaGG Uaaaaa@3D58@  The y-axis goes from 0 to 50 and the x-axis goes from 0.0 to 1.0. The scatter plot is showing an increasing linear relationship between x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpepeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuj0lXxdrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWG4baaaa@394C@  and y. MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpepeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuj0lXxdrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWG5bGaai Olaaaa@39FF@  The variation increases when x increases.

Figure 5.3 of article 14541

Description of Figure 5.3

Figure made of two scatter plots ( y MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaqabaqaai aadMhaaiaawIcaaaaa@3813@  versus x ), MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaqacaqaai aadIhaaiaawMcaaiaacYcaaaa@38C4@  each one illustrating an artificial population. The first graph is the population generated from y i =10 x i 1/4 + ε i , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWG5bWaaS baaSqaaiaadMgaaeqaaOGaeyypa0JaaGymaiaaicdacaWG4bWaa0ba aSqaaiaadMgaaeaadaWcgaqaaiaaigdaaeaacaaI0aaaaaaakiabgU caRiabew7aLnaaBaaaleaacaWGPbaabeaakiaacYcaaaa@42FA@  where ε i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH1oqzda WgaaWcbaGaamyAaaqabaGccqWI8iIoaaa@3A43@  i.i.d. Student t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWG0baaaa@3748@  with ν=5. MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH9oGBcq GH9aqpcaaI1aGaaiOlaaaa@3A7E@  The y-axis goes from 0 to 15 and the x-axis goes from 0.0 to 1.0. The scatter plot is showing an increasing concave relationship between x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpepeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuj0lXxdrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWG4baaaa@394C@  and y. MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpepeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuj0lXxdrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWG5bGaai Olaaaa@39FF@  The second graph is the population generated from y i = ε i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWG5bWaaS baaSqaaiaadMgaaeqaaOGaeyypa0JaeqyTdu2aaSbaaSqaaiaadMga aeqaaaaa@3C38@  and ε i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH1oqzda WgaaWcbaGaamyAaaqabaGccqWI8iIoaaa@3A43@  indep. noncentral Student t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWG0baaaa@3748@  with ν=5 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH9oGBcq GH9aqpcaaI1aaaaa@39CC@  and μ=15 x i . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH8oqBcq GH9aqpcaaIXaGaaGynaiaadIhadaWgaaWcbaGaamyAaaqabaGccaGG Uaaaaa@3D58@  The y-axis goes from 0 to 50 and the x-axis goes from 0.0 to 1.0. The scatter plot is showing an increasing concave relationship between x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpepeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuj0lXxdrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWG4baaaa@394C@  and y. MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpepeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuj0lXxdrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWG5bGaai Olaaaa@39FF@  The variation increases when x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpepeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuj0lXxdrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWG4baaaa@394C@  increases.

Figure 5.4 of article 14541

Description of Figure 5.4

Figure made of two scatter plots ( y MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaqabaqaai aadMhaaiaawIcaaaaa@3813@  versus x ), MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaqacaqaai aadIhaaiaawMcaaiaacYcaaaa@38C4@  each one illustrating a real population, MU284 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGnbGaam yvaiaaikdacaaI4aGaaGinaaaa@3A37@  Population of Sweden Municipalities of Särndal et al (1992). On the first graph, y i =lnRMT 85 i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWG5bWaaS baaSqaaiaadMgaaeqaaOGaeyypa0JaciiBaiaac6gacaWGsbGaamyt aiaadsfacaaI4aGaaGynamaaBaaaleaacaWGPbaabeaaaaa@4078@  for the i th MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGPbWaaW baaSqabeaacaqG0bGaaeiAaaaaaaa@394C@  municipality and x i =lnP 85 i . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWG4bWaaS baaSqaaiaadMgaaeqaaOGaeyypa0JaciiBaiaac6gacaWGqbGaaGio aiaaiwdadaWgaaWcbaGaamyAaaqabaGccaGGUaaaaa@3F86@  The y-axis goes from 3 to 9 and the x-axis goes from 1 to 6. The scatter plot is showing an increasing linear relationship between x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpepeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuj0lXxdrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWG4baaaa@394C@  and y. MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpepeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuj0lXxdrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWG5bGaai Olaaaa@39FF@  On the second graph, y i =lnRMT 85 i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWG5bWaaS baaSqaaiaadMgaaeqaaOGaeyypa0JaciiBaiaac6gacaWGsbGaamyt aiaadsfacaaI4aGaaGynamaaBaaaleaacaWGPbaabeaaaaa@4078@  for the i th MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGPbWaaW baaSqabeaacaqG0bGaaeiAaaaaaaa@394C@  municipality and x i =lnREV 84 i . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWG4bWaaS baaSqaaiaadMgaaeqaaOGaeyypa0JaciiBaiaac6gacaWGsbGaamyr aiaadAfacaaI4aGaaGinamaaBaaaleaacaWGPbaabeaakiaac6caaa a@412C@  The y-axis goes from 3 to 9 and the x-axis goes from 6 to 11. The scatter plot is showing a more variable increasing linear relationship between x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpepeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuj0lXxdrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWG4baaaa@394C@  and y. MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpepeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuj0lXxdrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWG5bGaai Olaaaa@39FF@

From each population we selected independently B = 1,000 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOqaiaai2 dacaqGXaGaaeilaiaabcdacaqGWaGaaeimaaaa@3A09@ samples. When sampling from the artificial populations we set the sample size equal to n = 100 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaiaai2 dacaaIXaGaaGimaiaaicdaaaa@38E8@ in case of simple random without replacement sampling and, in case of Poisson sampling, we set the expected sample size equal to n * = 100 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBamaaCa aaleqabaGaaGOkaaaakiaai2dacaaIXaGaaGimaiaaicdaaaa@39D3@ and made the sample inclusion probabilities proportional to the standard deviations of the shifted noncentral Student  t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiDaaaa@35F8@ distributions of above. When sampling from the real populations, we set the sample size equal to n = 30 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaiaai2 dacaaIZaGaaGimaaaa@3830@ in case of simple random without replacement sampling. In case of Poisson sampling, we set the expected sample size equal to n * = 30 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBamaaCa aaleqabaGaaGOkaaaakiaai2dacaaIZaGaaGimaaaa@391B@ and made the sample inclusion probabilities proportional to the absolute values of the residuals from the linear least squares regressions of the population y i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEamaaBa aaleaacaWGPbaabeaaaaa@3717@ values on the population x i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBa aaleaacaWGPbaabeaaaaa@3716@ values.

As for the definition of the nonparametric estimators, we used the Epanechnikov kernel function K ( u ) := 0.75 ( 1 u 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4samaabm aabaGaamyDaaGaayjkaiaawMcaaiaaiQdacaaI9aGaaGimaiaai6ca caaI3aGaaGynamaabmaabaGaaGymaiabgkHiTiaadwhadaahaaWcbe qaaiaaikdaaaaakiaawIcacaGLPaaaaaa@41ED@ with λ = 0.15 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4UdWMaaG ypaiaaicdacaaIUaGaaGymaiaaiwdaaaa@3A66@ or λ = 0.3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4UdWMaaG ypaiaaicdacaaIUaGaaG4maaaa@39A9@ for the samples taken from the artificial populations, and the Gaussian kernel function K ( u ) := 1 / 2 π e ( 1 / 2 ) u 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4samaabm aabaGaamyDaaGaayjkaiaawMcaaiaaiQdacaaI9aWaaSGbaeaacaaI XaaabaWaaOaaaeaacaaIYaGaeqiWdahaleqaaaaakiaadwgadaahaa WcbeqaaiabgkHiTmaabmaabaWaaSGbaeaacaaIXaaabaGaaGOmaaaa aiaawIcacaGLPaaacaWG1bWaaWbaaWqabeaacaaIYaaaaaaaaaa@444A@ with λ = 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4UdWMaaG ypaiaaigdaaaa@3835@ or λ = 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4UdWMaaG ypaiaaikdaaaa@3836@ for the samples taken from the real populations. In the tables with the simulation results the nonparametric estimators corresponding to the small and large bandwidth values are identified with an s MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4Caaaa@35F7@ (small) or an l MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiBaaaa@35F0@ (large) in the subscript. We resorted to the Gaussian kernel function for the samples taken from the real populations to avoid singularity problems that occur in case of holes in the sampled set of x i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBa aaleaacaWGPbaabeaakiabgkHiTaaa@380D@ values. Such holes are much more likely to occur with the real populations than with the artificial ones, because the distributions of the auxiliary variables are asymmetric in the former. In fact, in the artificial populations the nonparametric estimators were well-defined for all the B = 1,000 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOqaiaai2 dacaqGXaGaaeilaiaabcdacaqGWaGaaeimaaaa@3A09@ samples selected according to the simple random without replacement sampling design. For the Poisson sampling design, on the other hand, 47 among the B = 1,000 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOqaiaai2 dacaqGXaGaaeilaiaabcdacaqGWaGaaeimaaaa@3A09@ simulated samples were such that the nonparametric estimators with the small bandwidth value could not be computed and just one of these samples was such that the nonparametric estimators with the large bandwidth value were undefined. The simulation results referring to the nonparametric estimators in Tables 5.2 and 5.5 account only for the samples where they were well-defined and thus they are based on a little less than B = 1,000 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOqaiaai2 dacaqGXaGaaeilaiaabcdacaqGWaGaaeimaaaa@3A09@ realizations.

Tables 5.1 to 5.4 report the simulated bias (BIAS) and the simulated root mean square error (RMSE) for each distribution function estimator at different levels of t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiDaaaa@35F8@ at which F N ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOramaaBa aaleaacaWGobaabeaakmaabmaabaGaamiDaaGaayjkaiaawMcaaaaa @3955@ has been estimated: based, for example, on the values F ˜ b ( t ) , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaia WaaSbaaSqaaiaadkgaaeqaaOWaaeWaaeaacaWG0baacaGLOaGaayzk aaGaaiilaaaa@3A28@ b = 1,2, , B , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOyaiaai2 dacaaIXaGaaGilaiaaikdacaaISaGaeSOjGSKaaGilaiaadkeacaGG Saaaaa@3CDF@ taken on by the estimator F ˜ ( t ) , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaia WaaeWaaeaacaWG0baacaGLOaGaayzkaaGaaiilaaaa@390B@

BIAS  := 1 B b = 1 B ( F ˜ b ( t ) F N ( t ) ) × 10,000 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeOqaiaabM eacaqGbbGaae4uaiaaiQdacaaI9aWaaSaaaeaacaaIXaaabaGaamOq aaaadaaeWbqabSqaaiaadkgacaaI9aGaaGymaaqaaiaadkeaa0Gaey yeIuoakmaabmaabaGabmOrayaaiaWaaSbaaSqaaiaadkgaaeqaaOWa aeWaaeaacaWG0baacaGLOaGaayzkaaGaeyOeI0IaamOramaaBaaale aacaWGobaabeaakmaabmaabaGaamiDaaGaayjkaiaawMcaaaGaayjk aiaawMcaaiabgEna0kaabgdacaqGWaGaaeilaiaabcdacaqGWaGaae imaaaa@524E@

and

RMSE  : = 1 B b = 1 B ( F ˜ b ( t ) F N ( t ) ) 2 × 10,000 . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeOuaiaab2 eacaqGtbGaaeyraiaaiQdacaaI9aWaaOaaaeaadaWcaaqaaiaaigda aeaacaWGcbaaamaaqahabeWcbaGaamOyaiaai2dacaaIXaaabaGaam OqaaqdcqGHris5aOWaaeWaaeaaceWGgbGbaGaadaWgaaWcbaGaamOy aaqabaGcdaqadaqaaiaadshaaiaawIcacaGLPaaacqGHsislcaWGgb WaaSbaaSqaaiaad6eaaeqaaOWaaeWaaeaacaWG0baacaGLOaGaayzk aaaacaGLOaGaayzkaaWaaWbaaSqabeaacaaIYaaaaaqabaGccqGHxd aTcaqGXaGaaeimaiaabYcacaqGWaGaaeimaiaabcdacaqGUaaaaa@541A@

The RMSE’s show that the estimators based on the modified fitted values are usually more efficient. In sampling from the real populations the gain in RMSE is sometimes quite large. As expected, the model-based estimators tend to be more efficient than the generalized difference estimators in case of simple random without replacement sampling when both types of estimator are approximately unbiased. Under the Poisson sampling scheme the BIAS of the model-based estimators increases, but nonetheless they remain competitive. More variability in the sample inclusion probabilities would certainly change this outcome, because it would increase the BIAS of the model-based estimators. The simulation results should therefore not be seen to be in contrast with Johnson, Breidt and Opsomer (2008) who argue in favor of generalized difference estimators (called model-assisted estimators in their paper) as “a good overall choice for distribution function estimators”.

Table 5.1
Artificial populations (population size N = 1,000 ) . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbeqabeWacmGabiqabeqabmqabeabbaGcbaWaaeGaaeaaca WGobGaaGypaiaabgdacaqGSaGaaeimaiaabcdacaqGWaaacaGLPaaa caGGUaaaaa@3B88@ BIAS and RMSE of distribution function estimators under simple random without replacement sampling. Sample size n = 100 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbeqabeWacmGabiqabeqabmqabeabbaGcbaGaamOBaiaai2 dacaqGXaGaaeimaiaabcdaaaa@38CC@
Table summary
This table displays the results of Artificial populations (population size XXXX BIAS and RMSE of distribution function estimators under simple random without replacement sampling. Sample size XXXX XXXX , XXXX, BIAS , RMSE and RMSE, calculated using XXXX with XXXX i.i.d. central Student XXXX with XXXX, XXXX with XXXX indep. noncentral Student XXXX with XXXX and XXXX, XXXX with XXXX i.i.d. Student XXXX with XXXX, XXXX with XXXX indep. noncentral Student XXXX with XXXX and XXXX, XXXX with XXXX i.i.d. Student XXXX with XXXX and XXXX with XXXX indep. noncentral Student XXXX with XXXX and XXXX units of measure (appearing as column headers).
  t = F N 1 ( 0.05 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbeqabeWacmGabiqabeqabmqabeabbaGcbaGaamiDaiaai2 dacaWGgbWaa0baaSqaaiaad6eaaeaacqGHsislcaaIXaaaaOWaaeWa aeaacaaIWaGaaGOlaiaaicdacaaI1aaacaGLOaGaayzkaaaaaa@40DB@ t = F N 1 ( 0.25 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbeqabeWacmGabiqabeqabmqabeabbaGcbaGaamiDaiaai2 dacaWGgbWaa0baaSqaaiaad6eaaeaacqGHsislcaaIXaaaaOWaaeWa aeaacaaIWaGaaGOlaiaaikdacaaI1aaacaGLOaGaayzkaaaaaa@40DD@ t = F N 1 ( 0.50 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbeqabeWacmGabiqabeqabmqabeabbaGcbaGaamiDaiaai2 dacaWGgbWaa0baaSqaaiaad6eaaeaacqGHsislcaaIXaaaaOWaaeWa aeaacaaIWaGaaGOlaiaaiwdacaaIWaaacaGLOaGaayzkaaaaaa@40DB@ t = F N 1 ( 0.75 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbeqabeWacmGabiqabeqabmqabeabbaGcbaGaamiDaiaai2 dacaWGgbWaa0baaSqaaiaad6eaaeaacqGHsislcaaIXaaaaOWaaeWa aeaacaaIWaGaaGOlaiaaiEdacaaI1aaacaGLOaGaayzkaaaaaa@40E2@ t = F N 1 ( 0.95 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbeqabeWacmGabiqabeqabmqabeabbaGcbaGaamiDaiaai2 dacaWGgbWaa0baaSqaaiaad6eaaeaacqGHsislcaaIXaaaaOWaaeWa aeaacaaIWaGaaGOlaiaaiMdacaaI1aaacaGLOaGaayzkaaaaaa@40E4@
BIAS RMSE BIAS RMSE BIAS RMSE BIAS RMSE BIAS RMSE
y i = ε i , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEamaaBa aaleaacaWGPbaabeaakiaai2dacqaH1oqzdaWgaaWcbaGaamyAaaqa baGccaGGSaaaaa@3D84@ with ε i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyTdu2aaS baaSqaaiaadMgaaeqaaOGaeSipIOdaaa@3B14@ i.i.d. central Student t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiDaaaa@3819@ with ν = 5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyVd4MaaG ypaiaaiwdaaaa@3A5E@
F ^ s ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaja WaaSbaaSqaaiaadohaaeqaaOWaaeWaaeaacaWG0baacaGLOaGaayzk aaaaaa@3BAB@ 6 216 -3 433 31 512 23 434 12 207
F ^ l ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaja WaaSbaaSqaaiaadYgaaeqaaOWaaeWaaeaacaWG0baacaGLOaGaayzk aaaaaa@3BA4@ 15 219 10 430 0 502 -10 429 3 213
F ^ s * ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaja Waa0baaSqaaiaadohaaeaacaaIQaaaaOWaaeWaaeaacaWG0baacaGL OaGaayzkaaaaaa@3C60@ 6 209 -30 411 22 484 22 414 3 200
F ^ l * ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaja Waa0baaSqaaiaadYgaaeaacaaIQaaaaOWaaeWaaeaacaWG0baacaGL OaGaayzkaaaaaa@3C59@ 15 214 -9 409 10 477 1 407 -10 207
F ˜ s ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaia WaaSbaaSqaaiaadohaaeqaaOWaaeWaaeaacaWG0baacaGLOaGaayzk aaaaaa@3BAA@ 6 213 8 425 24 504 -4 430 8 207
F ˜ l ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaia WaaSbaaSqaaiaadYgaaeqaaOWaaeWaaeaacaWG0baacaGLOaGaayzk aaaaaa@3BA3@ 6 210 10 417 22 494 -8 422 6 206
F ˜ s * ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaia Waa0baaSqaaiaadohaaeaacaaIQaaaaOWaaeWaaeaacaWG0baacaGL OaGaayzkaaaaaa@3C5F@ 8 213 9 426 25 503 -5 432 5 206
F ˜ l * ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaia Waa0baaSqaaiaadYgaaeaacaaIQaaaaOWaaeWaaeaacaWG0baacaGL OaGaayzkaaaaaa@3C58@ 7 210 10 417 23 494 -6 424 4 206
F ˜ π ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaia WaaSbaaSqaaiabec8aWbqabaGcdaqadaqaaiaadshaaiaawIcacaGL Paaaaaa@3C6F@ 7 208 11 411 19 489 -5 417 6 200
  y i = ε i , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEamaaBa aaleaacaWGPbaabeaakiaai2dacqaH1oqzdaWgaaWcbaGaamyAaaqa baGccaGGSaaaaa@3D84@ with ε i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyTdu2aaS baaSqaaiaadMgaaeqaaOGaeSipIOdaaa@3B14@ indep. noncentral Student t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiDaaaa@3819@ with ν = 5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyVd4MaaG ypaiaaiwdaaaa@3A5E@ and μ = 15 x i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiVd0MaaG ypaiaaigdacaaI1aGaamiEamaaBaaaleaacaWGPbaabeaaaaa@3D2E@
F ^ s ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaja WaaSbaaSqaaiaadohaaeqaaOWaaeWaaeaacaWG0baacaGLOaGaayzk aaaaaa@3BAB@ 26 225 33 376 8 477 26 419 33 209
F ^ l ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaja WaaSbaaSqaaiaadYgaaeqaaOWaaeWaaeaacaWG0baacaGLOaGaayzk aaaaaa@3BA4@ 52 236 23 374 -5 475 38 421 29 213
F ^ s * ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaja Waa0baaSqaaiaadohaaeaacaaIQaaaaOWaaeWaaeaacaWG0baacaGL OaGaayzkaaaaaa@3C60@ 20 195 -29 351 -89 471 11 407 30 202
F ^ l * ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaja Waa0baaSqaaiaadYgaaeaacaaIQaaaaOWaaeWaaeaacaWG0baacaGL OaGaayzkaaaaaa@3C59@ 36 201 -11 357 -94 473 28 410 21 204
F ˜ s ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaia WaaSbaaSqaaiaadohaaeqaaOWaaeWaaeaacaWG0baacaGLOaGaayzk aaaaaa@3BAA@ 8 211 11 370 -7 473 4 415 16 211
F ˜ l ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaia WaaSbaaSqaaiaadYgaaeqaaOWaaeWaaeaacaWG0baacaGLOaGaayzk aaaaaa@3BA3@ 5 208 8 367 -5 468 5 411 16 212
F ˜ s * ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaia Waa0baaSqaaiaadohaaeaacaaIQaaaaOWaaeWaaeaacaWG0baacaGL OaGaayzkaaaaaa@3C5F@ 11 210 11 372 -11 475 4 416 15 210
F ˜ l * ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaia Waa0baaSqaaiaadYgaaeaacaaIQaaaaOWaaeWaaeaacaWG0baacaGL OaGaayzkaaaaaa@3C58@ 7 208 11 368 -7 468 8 412 15 211
F ˜ π ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaia WaaSbaaSqaaiabec8aWbqabaGcdaqadaqaaiaadshaaiaawIcacaGL Paaaaaa@3C6F@ 1 211 1 391 -6 477 8 399 18 210
  y i = 10 x i + ε i , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEamaaBa aaleaacaWGPbaabeaakiaai2dacaaIXaGaaGimaiaadIhadaWgaaWc baGaamyAaaqabaGccqGHRaWkcqaH1oqzdaWgaaWcbaGaamyAaaqaba GccaGGSaaaaa@41FC@ with ε i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyTdu2aaS baaSqaaiaadMgaaeqaaOGaeSipIOdaaa@3B14@ i.i.d. Student t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiDaaaa@3819@ with ν = 5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyVd4MaaG ypaiaaiwdaaaa@3A5E@
F ^ s ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaja WaaSbaaSqaaiaadohaaeqaaOWaaeWaaeaacaWG0baacaGLOaGaayzk aaaaaa@3BAB@ 32 201 25 275 13 250 -14 264 -36 217
F ^ l ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaja WaaSbaaSqaaiaadYgaaeqaaOWaaeWaaeaacaWG0baacaGLOaGaayzk aaaaaa@3BA4@ 114 250 152 304 12 236 -180 312 -86 242
F ^ s * ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaja Waa0baaSqaaiaadohaaeaacaaIQaaaaOWaaeWaaeaacaWG0baacaGL OaGaayzkaaaaaa@3C60@ -50 165 12 226 51 216 26 230 13 172
F ^ l * ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaja Waa0baaSqaaiaadYgaaeaacaaIQaaaaOWaaeWaaeaacaWG0baacaGL OaGaayzkaaaaaa@3C59@ -46 155 -14 199 69 195 23 211 17 156
F ˜ s ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaia WaaSbaaSqaaiaadohaaeqaaOWaaeWaaeaacaWG0baacaGLOaGaayzk aaaaaa@3BAA@ -5 186 4 275 15 248 11 269 -2 201
F ˜ l ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaia WaaSbaaSqaaiaadYgaaeqaaOWaaeWaaeaacaWG0baacaGLOaGaayzk aaaaaa@3BA3@ -5 184 7 274 17 250 5 269 -2 196
F ˜ s * ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaia Waa0baaSqaaiaadohaaeaacaaIQaaaaOWaaeWaaeaacaWG0baacaGL OaGaayzkaaaaaa@3C5F@ -10 180 5 275 16 245 14 266 -1 200
F ˜ l * ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaia Waa0baaSqaaiaadYgaaeaacaaIQaaaaOWaaeWaaeaacaWG0baacaGL OaGaayzkaaaaaa@3C58@ -9 176 3 272 15 242 13 262 -1 194
F ˜ π ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaia WaaSbaaSqaaiabec8aWbqabaGcdaqadaqaaiaadshaaiaawIcacaGL Paaaaaa@3C6F@ -7 203 14 413 37 472 17 405 1 206
  y i = 10 x i + ε i , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEamaaBa aaleaacaWGPbaabeaakiaai2dacaaIXaGaaGimaiaadIhadaWgaaWc baGaamyAaaqabaGccqGHRaWkcqaH1oqzdaWgaaWcbaGaamyAaaqaba GccaGGSaaaaa@41FC@ with ε i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyTdu2aaS baaSqaaiaadMgaaeqaaOGaeSipIOdaaa@3B14@ indep. noncentral Student t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiDaaaa@3819@ with ν = 5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyVd4MaaG ypaiaaiwdaaaa@3A5E@ and μ = 15 x i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiVd0MaaG ypaiaaigdacaaI1aGaamiEamaaBaaaleaacaWGPbaabeaaaaa@3D2E@
F ^ s ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaja WaaSbaaSqaaiaadohaaeqaaOWaaeWaaeaacaWG0baacaGLOaGaayzk aaaaaa@3BAB@ 24 204 23 351 27 403 26 382 29 208
F ^ l ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaja WaaSbaaSqaaiaadYgaaeqaaOWaaeWaaeaacaWG0baacaGLOaGaayzk aaaaaa@3BA4@ 94 242 135 372 51 392 13 380 15 212
F ^ s * ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaja Waa0baaSqaaiaadohaaeaacaaIQaaaaOWaaeWaaeaacaWG0baacaGL OaGaayzkaaaaaa@3C60@ 55 182 -9 301 -18 368 -23 359 37 202
F ^ l * ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaja Waa0baaSqaaiaadYgaaeaacaaIQaaaaOWaaeWaaeaacaWG0baacaGL OaGaayzkaaaaaa@3C59@ 124 210 -31 278 -63 363 -8 356 48 200
F ˜ s ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaia WaaSbaaSqaaiaadohaaeqaaOWaaeWaaeaacaWG0baacaGLOaGaayzk aaaaaa@3BAA@ -2 194 -4 349 11 401 18 377 13 208
F ˜ l ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaia WaaSbaaSqaaiaadYgaaeqaaOWaaeWaaeaacaWG0baacaGLOaGaayzk aaaaaa@3BA3@ -2 190 -5 345 12 398 17 374 11 209
F ˜ s * ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaia Waa0baaSqaaiaadohaaeaacaaIQaaaaOWaaeWaaeaacaWG0baacaGL OaGaayzkaaaaaa@3C5F@ 0 191 -5 352 14 401 20 376 13 207
F ˜ l * ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaia Waa0baaSqaaiaadYgaaeaacaaIQaaaaOWaaeWaaeaacaWG0baacaGL OaGaayzkaaaaaa@3C58@ -1 189 -6 344 13 397 18 375 12 209
F ˜ π ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaia WaaSbaaSqaaiabec8aWbqabaGcdaqadaqaaiaadshaaiaawIcacaGL Paaaaaa@3C6F@ -4 205 -5 401 21 470 24 401 14 207
  y i = 10 x i 1 / 4 + ε i , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEamaaBa aaleaacaWGPbaabeaakiaai2dacaaIXaGaaGimaiaadIhadaqhaaWc baGaamyAaaqaamaalyaabaGaaGymaaqaaiaaisdaaaaaaOGaey4kaS IaeqyTdu2aaSbaaSqaaiaadMgaaeqaaOGaaiilaaaa@438C@ with ε i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyTdu2aaS baaSqaaiaadMgaaeqaaOGaeSipIOdaaa@3B14@ i.i.d. Student t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiDaaaa@3819@ with ν = 5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyVd4MaaG ypaiaaiwdaaaa@3A5E@
F ^ s ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaja WaaSbaaSqaaiaadohaaeqaaOWaaeWaaeaacaWG0baacaGLOaGaayzk aaaaaa@3BAB@ 81 207 44 316 17 384 -2 376 23 203
F ^ l ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaja WaaSbaaSqaaiaadYgaaeqaaOWaaeWaaeaacaWG0baacaGLOaGaayzk aaaaaa@3BA4@ 138 258 183 356 35 367 -50 374 8 208
F ^ s * ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaja Waa0baaSqaaiaadohaaeaacaaIQaaaaOWaaeWaaeaacaWG0baacaGL OaGaayzkaaaaaa@3C60@ 7 146 -14 274 16 352 -8 358 15 197
F ^ l * ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaja Waa0baaSqaaiaadYgaaeaacaaIQaaaaOWaaeWaaeaacaWG0baacaGL OaGaayzkaaaaaa@3C59@ 9 144 10 246 -2 323 -18 339 24 186
F ˜ s ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaia WaaSbaaSqaaiaadohaaeqaaOWaaeWaaeaacaWG0baacaGLOaGaayzk aaaaaa@3BAA@ 3 175 3 319 10 383 17 374 10 203
F ˜ l ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaia WaaSbaaSqaaiaadYgaaeqaaOWaaeWaaeaacaWG0baacaGLOaGaayzk aaaaaa@3BA3@ 0 178 5 316 11 380 17 370 8 202
F ˜ s * ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaia Waa0baaSqaaiaadohaaeaacaaIQaaaaOWaaeWaaeaacaWG0baacaGL OaGaayzkaaaaaa@3C5F@ 1 167 5 320 12 383 17 374 9 203
F ˜ l * ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaia Waa0baaSqaaiaadYgaaeaacaaIQaaaaOWaaeWaaeaacaWG0baacaGL OaGaayzkaaaaaa@3C58@ -1 164 6 316 13 379 20 368 8 201
F ˜ π ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaia WaaSbaaSqaaiabec8aWbqabaGcdaqadaqaaiaadshaaiaawIcacaGL Paaaaaa@3C6F@ 4 209 11 412 25 477 27 422 10 200
  y i = 10 x i 1 / 4 + ε i , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEamaaBa aaleaacaWGPbaabeaakiaai2dacaaIXaGaaGimaiaadIhadaqhaaWc baGaamyAaaqaamaalyaabaGaaGymaaqaaiaaisdaaaaaaOGaey4kaS IaeqyTdu2aaSbaaSqaaiaadMgaaeqaaOGaaiilaaaa@438C@ with ε i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyTdu2aaS baaSqaaiaadMgaaeqaaOGaeSipIOdaaa@3B14@ indep. noncentral Student t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiDaaaa@3819@ with ν = 5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyVd4MaaG ypaiaaiwdaaaa@3A5E@ and μ = 15 x i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiVd0MaaG ypaiaaigdacaaI1aGaamiEamaaBaaaleaacaWGPbaabeaaaaa@3D2E@
F ^ s ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaja WaaSbaaSqaaiaadohaaeqaaOWaaeWaaeaacaWG0baacaGLOaGaayzk aaaaaa@3BAB@ 59 234 95 402 66 455 51 395 26 208
F ^ l ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaja WaaSbaaSqaaiaadYgaaeqaaOWaaeWaaeaacaWG0baacaGLOaGaayzk aaaaaa@3BA4@ 94 259 190 441 147 467 98 400 16 212
F ^ s * ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaja Waa0baaSqaaiaadohaaeaacaaIQaaaaOWaaeWaaeaacaWG0baacaGL OaGaayzkaaaaaa@3C60@ 30 184 33 343 -123 435 -34 385 40 203
F ^ l * ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaja Waa0baaSqaaiaadYgaaeaacaaIQaaaaOWaaeWaaeaacaWG0baacaGL OaGaayzkaaaaaa@3C59@ 57 201 58 331 -148 437 2 382 34 203
F ˜ s ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaia WaaSbaaSqaaiaadohaaeqaaOWaaeWaaeaacaWG0baacaGLOaGaayzk aaaaaa@3BAA@ 1 205 7 386 12 449 17 392 13 208
F ˜ l ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaia WaaSbaaSqaaiaadYgaaeqaaOWaaeWaaeaacaWG0baacaGLOaGaayzk aaaaaa@3BA3@ -1 204 0 385 9 445 20 389 11 209
F ˜ s * ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaia Waa0baaSqaaiaadohaaeaacaaIQaaaaOWaaeWaaeaacaWG0baacaGL OaGaayzkaaaaaa@3C5F@ 3 201 8 389 7 449 13 392 14 207
F ˜ l * ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaia Waa0baaSqaaiaadYgaaeaacaaIQaaaaOWaaeWaaeaacaWG0baacaGL OaGaayzkaaaaaa@3C58@ 0 198 6 383 9 446 19 390 13 208
F ˜ π ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaia WaaSbaaSqaaiabec8aWbqabaGcdaqadaqaaiaadshaaiaawIcacaGL Paaaaaa@3C6F@ 0 205 -2 399 9 463 25 398 14 208
Table 5.2
Artificial populations (population size N = 1,000 ) . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbeqabeWacmGabiqabeqabmqabeabbaGcbaWaaeGaaeaaca WGobGaaGypaiaabgdacaqGSaGaaeimaiaabcdacaqGWaaacaGLPaaa caGGUaaaaa@3B88@ BIAS and RMSE of distribution function estimators under Poisson sampling with sample inclusion probabilities π i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbeqabeWacmGabiqabeqabmqabeabbaGcbaGaeqiWda3aaS baaSqaaiaadMgaaeqaaaaa@37CF@ proportional to the standard deviations of the noncentral Student t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbeqabeWacmGabiqabeqabmqabeabbaGcbaGaamiDaaaa@35F1@ distributions with ν = 5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbeqabeWacmGabiqabeqabmqabeabbaGcbaGaeqyVd4MaaG ypaiaaiwdaaaa@3836@ d.o.f. and with noncentrality parameters μ = 15 x i . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbeqabeWacmGabiqabeqabmqabeabbaGcbaGaeqiVd0MaaG ypaiaaigdacaaI1aGaamiEamaaBaaaleaacaWGPbaabeaakiaac6ca aaa@3BC2@ Expected sample size n * = 100 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbeqabeWacmGabiqabeqabmqabeabbaGcbaGaamOBamaaCa aaleqabaGaaGOkaaaakiaai2dacaaIXaGaaGimaiaaicdaaaa@39CC@
Table summary
This table displays the results of Artificial populations (population size XXXX BIAS and RMSE of distribution function estimators under Poisson sampling with sample inclusion probabilities XXXX proportional to the standard deviations of the noncentral Student XXXX distributions with XXXX d.o.f. and with noncentrality parameters XXXX Expected sample size XXXX XXXX, BIAS , RMSE and RMSE, calculated using XXXX with XXXX i.i.d. central Student XXXX with XXXX, XXXX with XXXX indep. noncentral Student XXXX with XXXX and XXXX, XXXX with XXXX i.i.d. Student XXXX with XXXX, XXXX with XXXX indep. noncentral Student XXXX with XXXX and XXXX, XXXX with XXXX i.i.d. Student XXXX with XXXX and XXXX with XXXX indep. noncentral Student XXXX with XXXX and XXXX units of measure (appearing as column headers).
  t = F N 1 ( 0.05 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbeqabeWacmGabiqabeqabmqabeabbaGcbaGaamiDaiaai2 dacaWGgbWaa0baaSqaaiaad6eaaeaacqGHsislcaaIXaaaaOWaaeWa aeaacaaIWaGaaGOlaiaaicdacaaI1aaacaGLOaGaayzkaaaaaa@40DB@ t = F N 1 ( 0.25 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbeqabeWacmGabiqabeqabmqabeabbaGcbaGaamiDaiaai2 dacaWGgbWaa0baaSqaaiaad6eaaeaacqGHsislcaaIXaaaaOWaaeWa aeaacaaIWaGaaGOlaiaaikdacaaI1aaacaGLOaGaayzkaaaaaa@40DD@ t = F N 1 ( 0.50 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbeqabeWacmGabiqabeqabmqabeabbaGcbaGaamiDaiaai2 dacaWGgbWaa0baaSqaaiaad6eaaeaacqGHsislcaaIXaaaaOWaaeWa aeaacaaIWaGaaGOlaiaaiwdacaaIWaaacaGLOaGaayzkaaaaaa@40DB@ t = F N 1 ( 0.75 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbeqabeWacmGabiqabeqabmqabeabbaGcbaGaamiDaiaai2 dacaWGgbWaa0baaSqaaiaad6eaaeaacqGHsislcaaIXaaaaOWaaeWa aeaacaaIWaGaaGOlaiaaiEdacaaI1aaacaGLOaGaayzkaaaaaa@40E2@ t = F N 1 ( 0.95 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbeqabeWacmGabiqabeqabmqabeabbaGcbaGaamiDaiaai2 dacaWGgbWaa0baaSqaaiaad6eaaeaacqGHsislcaaIXaaaaOWaaeWa aeaacaaIWaGaaGOlaiaaiMdacaaI1aaacaGLOaGaayzkaaaaaa@40E4@
BIAS RMSE BIAS RMSE BIAS RMSE BIAS RMSE BIAS RMSE
y i = ε i , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEamaaBa aaleaacaWGPbaabeaakiaai2dacqaH1oqzdaWgaaWcbaGaamyAaaqa baGccaGGSaaaaa@3D84@ with ε i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyTdu2aaS baaSqaaiaadMgaaeqaaOGaeSipIOdaaa@3B14@ i.i.d. central Student t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiDaaaa@3819@ with ν = 5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyVd4MaaG ypaiaaiwdaaaa@3A5E@
F ^ s ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaja WaaSbaaSqaaiaadohaaeqaaOWaaeWaaeaacaWG0baacaGLOaGaayzk aaaaaa@3BAB@ -10 252 -11 593 -22 738 -20 743 6 357
F ^ l ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaja WaaSbaaSqaaiaadYgaaeqaaOWaaeWaaeaacaWG0baacaGLOaGaayzk aaaaaa@3BA4@ -1 237 9 543 -15 621 -5 590 11 302
F ^ s * ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaja Waa0baaSqaaiaadohaaeaacaaIQaaaaOWaaeWaaeaacaWG0baacaGL OaGaayzkaaaaaa@3C60@ 22 244 -29 485 -3 555 9 515 -17 297
F ^ l * ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaja Waa0baaSqaaiaadYgaaeaacaaIQaaaaOWaaeWaaeaacaWG0baacaGL OaGaayzkaaaaaa@3C59@ 14 238 -10 492 -5 564 14 524 -1 283
F ˜ s ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaia WaaSbaaSqaaiaadohaaeqaaOWaaeWaaeaacaWG0baacaGLOaGaayzk aaaaaa@3BAA@ -6 247 0 579 -27 724 -40 736 3 349
F ˜ l ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaia WaaSbaaSqaaiaadYgaaeqaaOWaaeWaaeaacaWG0baacaGLOaGaayzk aaaaaa@3BA3@ -2 231 11 526 -1 598 -10 566 7 285
F ˜ s * ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaia Waa0baaSqaaiaadohaaeaacaaIQaaaaOWaaeWaaeaacaWG0baacaGL OaGaayzkaaaaaa@3C5F@ 23 248 23 505 -4 562 -27 531 -20 304
F ˜ l * ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaia Waa0baaSqaaiaadYgaaeaacaaIQaaaaOWaaeWaaeaacaWG0baacaGL OaGaayzkaaaaaa@3C58@ 12 240 20 504 1 573 -13 538 -6 287
F ˜ π ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaia WaaSbaaSqaaiabec8aWbqabaGcdaqadaqaaiaadshaaiaawIcacaGL Paaaaaa@3C6F@ -6 220 -7 543 -37 741 -44 929 -48 1,058
  y i = ε i , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEamaaBa aaleaacaWGPbaabeaakiaai2dacqaH1oqzdaWgaaWcbaGaamyAaaqa baGccaGGSaaaaa@3D84@ with ε i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyTdu2aaS baaSqaaiaadMgaaeqaaOGaeSipIOdaaa@3B14@ indep. noncentral Student t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiDaaaa@3819@ with ν = 5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyVd4MaaG ypaiaaiwdaaaa@3A5E@ and μ = 15 x i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiVd0MaaG ypaiaaigdacaaI1aGaamiEamaaBaaaleaacaWGPbaabeaaaaa@3D2E@
F ^ s ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaja WaaSbaaSqaaiaadohaaeqaaOWaaeWaaeaacaWG0baacaGLOaGaayzk aaaaaa@3BAB@ 17 164 30 411 4 749 14 590 15 190
F ^ l ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaja WaaSbaaSqaaiaadYgaaeqaaOWaaeWaaeaacaWG0baacaGLOaGaayzk aaaaaa@3BA4@ 47 173 19 383 -1 602 57 498 15 187
F ^ s * ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaja Waa0baaSqaaiaadohaaeaacaaIQaaaaOWaaeWaaeaacaWG0baacaGL OaGaayzkaaaaaa@3C60@ 21 175 -7 378 -89 554 -11 473 3 192
F ^ l * ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaja Waa0baaSqaaiaadYgaaeaacaaIQaaaaOWaaeWaaeaacaWG0baacaGL OaGaayzkaaaaaa@3C59@ 29 152 -3 367 -99 555 27 481 3 184
F ˜ s ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaia WaaSbaaSqaaiaadohaaeqaaOWaaeWaaeaacaWG0baacaGLOaGaayzk aaaaaa@3BAA@ 1 159 10 406 -11 737 -5 579 -2 194
F ˜ l ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaia WaaSbaaSqaaiaadYgaaeqaaOWaaeWaaeaacaWG0baacaGLOaGaayzk aaaaaa@3BA3@ 1 158 9 388 -5 586 14 482 -1 192
F ˜ s * ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaia Waa0baaSqaaiaadohaaeaacaaIQaaaaOWaaeWaaeaacaWG0baacaGL OaGaayzkaaaaaa@3C5F@ 14 186 27 409 -3 562 -17 487 -10 200
F ˜ l * ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaia Waa0baaSqaaiaadYgaaeaacaaIQaaaaOWaaeWaaeaacaWG0baacaGL OaGaayzkaaaaaa@3C58@ 3 160 22 399 -11 566 -5 482 -2 193
F ˜ π ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaia WaaSbaaSqaaiabec8aWbqabaGcdaqadaqaaiaadshaaiaawIcacaGL Paaaaaa@3C6F@ -3 162 -7 451 -31 738 -29 980 -55 1,067
  y i = 10 x i + ε i , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEamaaBa aaleaacaWGPbaabeaakiaai2dacaaIXaGaaGimaiaadIhadaWgaaWc baGaamyAaaqabaGccqGHRaWkcqaH1oqzdaWgaaWcbaGaamyAaaqaba GccaGGSaaaaa@41FC@ with ε i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyTdu2aaS baaSqaaiaadMgaaeqaaOGaeSipIOdaaa@3B14@ i.i.d. Student t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiDaaaa@3819@ with ν = 5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyVd4MaaG ypaiaaiwdaaaa@3A5E@
F ^ s ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaja WaaSbaaSqaaiaadohaaeqaaOWaaeWaaeaacaWG0baacaGLOaGaayzk aaaaaa@3BAB@ 8 461 21 561 -12 259 -18 218 -30 164
F ^ l ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaja WaaSbaaSqaaiaadYgaaeqaaOWaaeWaaeaacaWG0baacaGLOaGaayzk aaaaaa@3BA4@ 78 429 183 451 2 248 -161 261 -79 189
F ^ s * ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaja Waa0baaSqaaiaadohaaeaacaaIQaaaaOWaaeWaaeaacaWG0baacaGL OaGaayzkaaaaaa@3C60@ -69 306 12 340 10 267 15 199 6 143
F ^ l * ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaja Waa0baaSqaaiaadYgaaeaacaaIQaaaaOWaaeWaaeaacaWG0baacaGL OaGaayzkaaaaaa@3C59@ -59 294 4 302 56 205 15 172 17 124
F ˜ s ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaia WaaSbaaSqaaiaadohaaeqaaOWaaeWaaeaacaWG0baacaGLOaGaayzk aaaaaa@3BAA@ -25 441 4 560 -10 257 9 219 5 153
F ˜ l ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaia WaaSbaaSqaaiaadYgaaeqaaOWaaeWaaeaacaWG0baacaGLOaGaayzk aaaaaa@3BA3@ -14 372 35 410 -10 262 4 219 5 151
F ˜ s * ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaia Waa0baaSqaaiaadohaaeaacaaIQaaaaOWaaeWaaeaacaWG0baacaGL OaGaayzkaaaaaa@3C5F@ -31 333 -2 386 -29 294 4 227 -1 161
F ˜ l * ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaia Waa0baaSqaaiaadYgaaeaacaaIQaaaaOWaaeWaaeaacaWG0baacaGL OaGaayzkaaaaaa@3C58@ -20 339 15 372 -10 259 11 215 4 151
F ˜ π ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaia WaaSbaaSqaaiabec8aWbqabaGcdaqadaqaaiaadshaaiaawIcacaGL Paaaaaa@3C6F@ -15 385 3 746 -37 917 -35 1,004 -48 1,070
  y i = 10 x i + ε i , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEamaaBa aaleaacaWGPbaabeaakiaai2dacaaIXaGaaGimaiaadIhadaWgaaWc baGaamyAaaqabaGccqGHRaWkcqaH1oqzdaWgaaWcbaGaamyAaaqaba GccaGGSaaaaa@41FC@ with ε i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyTdu2aaS baaSqaaiaadMgaaeqaaOGaeSipIOdaaa@3B14@ indep. noncentral Student t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiDaaaa@3819@ with ν = 5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyVd4MaaG ypaiaaiwdaaaa@3A5E@ and μ = 15 x i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiVd0MaaG ypaiaaigdacaaI1aGaamiEamaaBaaaleaacaWGPbaabeaaaaa@3D2E@
F ^ s ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaja WaaSbaaSqaaiaadohaaeqaaOWaaeWaaeaacaWG0baacaGLOaGaayzk aaaaaa@3BAB@ -4 516 30 671 7 453 11 344 6 182
F ^ l ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaja WaaSbaaSqaaiaadYgaaeqaaOWaaeWaaeaacaWG0baacaGLOaGaayzk aaaaaa@3BA4@ 63 409 129 539 61 421 9 341 1 180
F ^ s * ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaja Waa0baaSqaaiaadohaaeaacaaIQaaaaOWaaeWaaeaacaWG0baacaGL OaGaayzkaaaaaa@3C60@ 44 300 -29 433 -45 422 -47 345 12 180
F ^ l * ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaja Waa0baaSqaaiaadYgaaeaacaaIQaaaaOWaaeWaaeaacaWG0baacaGL OaGaayzkaaaaaa@3C59@ 107 314 -41 420 -60 397 -22 323 31 171
F ˜ s ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaia WaaSbaaSqaaiaadohaaeqaaOWaaeWaaeaacaWG0baacaGLOaGaayzk aaaaaa@3BAA@ -27 502 8 667 -8 450 0 344 -8 185
F ˜ l ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaia WaaSbaaSqaaiaadYgaaeqaaOWaaeWaaeaacaWG0baacaGLOaGaayzk aaaaaa@3BA3@ -10 364 16 510 11 425 -2 345 -7 182
F ˜ s * ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaia Waa0baaSqaaiaadohaaeaacaaIQaaaaOWaaeWaaeaacaWG0baacaGL OaGaayzkaaaaaa@3C5F@ -6 325 -9 479 -25 447 -14 356 -10 187
F ˜ l * ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaia Waa0baaSqaaiaadYgaaeaacaaIQaaaaOWaaeWaaeaacaWG0baacaGL OaGaayzkaaaaaa@3C58@ -7 332 -9 489 -5 426 -3 344 -6 182
F ˜ π ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaia WaaSbaaSqaaiabec8aWbqabaGcdaqadaqaaiaadshaaiaawIcacaGL Paaaaaa@3C6F@ -16 349 -2 705 -21 886 -42 1,013 -61 1,069
  y i = 10 x i 1 / 4 + ε i , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEamaaBa aaleaacaWGPbaabeaakiaai2dacaaIXaGaaGimaiaadIhadaqhaaWc baGaamyAaaqaamaalyaabaGaaGymaaqaaiaaisdaaaaaaOGaey4kaS IaeqyTdu2aaSbaaSqaaiaadMgaaeqaaOGaaiilaaaa@438C@ with ε i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyTdu2aaS baaSqaaiaadMgaaeqaaOGaeSipIOdaaa@3B14@ i.i.d. Student t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiDaaaa@3819@ with ν = 5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyVd4MaaG ypaiaaiwdaaaa@3A5E@
F ^ s ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaja WaaSbaaSqaaiaadohaaeqaaOWaaeWaaeaacaWG0baacaGLOaGaayzk aaaaaa@3BAB@ 36 497 47 629 9 418 -11 320 15 191
F ^ l ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaja WaaSbaaSqaaiaadYgaaeqaaOWaaeWaaeaacaWG0baacaGLOaGaayzk aaaaaa@3BA4@ 56 393 186 490 43 383 -48 308 13 184
F ^ s * ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaja Waa0baaSqaaiaadohaaeaacaaIQaaaaOWaaeWaaeaacaWG0baacaGL OaGaayzkaaaaaa@3C60@ -29 276 -19 383 -18 380 -43 335 -1 204
F ^ l * ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaja Waa0baaSqaaiaadYgaaeaacaaIQaaaaOWaaeWaaeaacaWG0baacaGL OaGaayzkaaaaaa@3C59@ -29 274 10 355 7 336 -29 290 23 179
F ˜ s ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaia WaaSbaaSqaaiaadohaaeqaaOWaaeWaaeaacaWG0baacaGLOaGaayzk aaaaaa@3BAA@ -30 475 12 630 4 421 7 317 6 191
F ˜ l ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaia WaaSbaaSqaaiaadYgaaeqaaOWaaeWaaeaacaWG0baacaGLOaGaayzk aaaaaa@3BA3@ -42 336 31 452 11 390 8 312 8 186
F ˜ s * ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaia Waa0baaSqaaiaadohaaeaacaaIQaaaaOWaaeWaaeaacaWG0baacaGL OaGaayzkaaaaaa@3C5F@ -31 306 5 429 -18 406 -14 344 -8 210
F ˜ l * ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaia Waa0baaSqaaiaadYgaaeaacaaIQaaaaOWaaeWaaeaacaWG0baacaGL OaGaayzkaaaaaa@3C58@ -28 308 14 424 7 387 5 315 7 191
F ˜ π ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaia WaaSbaaSqaaiabec8aWbqabaGcdaqadaqaaiaadshaaiaawIcacaGL Paaaaaa@3C6F@ -15 380 10 739 -23 891 -37 993 -47 1,064
  y i = 10 x i 1 / 4 + ε i , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEamaaBa aaleaacaWGPbaabeaakiaai2dacaaIXaGaaGimaiaadIhadaqhaaWc baGaamyAaaqaamaalyaabaGaaGymaaqaaiaaisdaaaaaaOGaey4kaS IaeqyTdu2aaSbaaSqaaiaadMgaaeqaaOGaaiilaaaa@438C@ with ε i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyTdu2aaS baaSqaaiaadMgaaeqaaOGaeSipIOdaaa@3B14@ indep. noncentral Student t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiDaaaa@3819@ with ν = 5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyVd4MaaG ypaiaaiwdaaaa@3A5E@ and μ = 15 x i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiVd0MaaG ypaiaaigdacaaI1aGaamiEamaaBaaaleaacaWGPbaabeaaaaa@3D2E@
F ^ s ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaja WaaSbaaSqaaiaadohaaeqaaOWaaeWaaeaacaWG0baacaGLOaGaayzk aaaaaa@3BAB@ 24 308 69 687 53 690 38 406 2 188
F ^ l ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaja WaaSbaaSqaaiaadYgaaeqaaOWaaeWaaeaacaWG0baacaGLOaGaayzk aaaaaa@3BA4@ 47 301 131 553 139 561 91 393 -2 186
F ^ s * ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaja Waa0baaSqaaiaadohaaeaacaaIQaaaaOWaaeWaaeaacaWG0baacaGL OaGaayzkaaaaaa@3C60@ 15 237 2 435 -135 513 -59 411 12 186
F ^ l * ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaja Waa0baaSqaaiaadYgaaeaacaaIQaaaaOWaaeWaaeaacaWG0baacaGL OaGaayzkaaaaaa@3C59@ 27 235 18 435 -149 506 -5 374 13 179
F ˜ s ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaia WaaSbaaSqaaiaadohaaeqaaOWaaeWaaeaacaWG0baacaGLOaGaayzk aaaaaa@3BAA@ -28 274 -8 673 4 688 3 403 -10 191
F ˜ l ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaia WaaSbaaSqaaiaadYgaaeqaaOWaaeWaaeaacaWG0baacaGLOaGaayzk aaaaaa@3BA3@ -29 251 -12 512 17 541 7 395 -9 188
F ˜ s * ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaia Waa0baaSqaaiaadohaaeaacaaIQaaaaOWaaeWaaeaacaWG0baacaGL OaGaayzkaaaaaa@3C5F@ -3 255 -12 481 -7 536 -20 422 -12 196
F ˜ l * ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaia Waa0baaSqaaiaadYgaaeaacaaIQaaaaOWaaeWaaeaacaWG0baacaGL OaGaayzkaaaaaa@3C58@ -12 251 -16 489 2 538 -4 399 -9 189
F ˜ π ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaia WaaSbaaSqaaiabec8aWbqabaGcdaqadaqaaiaadshaaiaawIcacaGL Paaaaaa@3C6F@ -10 267 -8 608 -4 860 -38 1,009 -63 1,066
Table 5.3
Real populations (population size N = 284 ) . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbeqabeWacmGabiqabeqabmqabeabbaGcbaWaaeGaaeaaca WGobGaaGypaiaaikdacaaI4aGaaGinaaGaayzkaaGaaiOlaaaa@3A48@ BIAS and RMSE of distribution function estimators under simple random without replacement sampling. Sample size n = 30 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbeqabeWacmGabiqabeqabmqabeabbaGcbaGaamOBaiaai2 dacaaIZaGaaGimaaaa@3829@
Table summary
This table displays the results of Real populations (population size XXXX BIAS and RMSE of distribution function estimators under simple random without replacement sampling. Sample size XXXX XXXX, BIAS , RMSE , RMSE and RBIAS , calculated using MU284 population with XXXX and XXXX units of measure (appearing as column headers).
  t = F N 1 ( 0.05 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbeqabeWacmGabiqabeqabmqabeabbaGcbaGaamiDaiaai2 dacaWGgbWaa0baaSqaaiaad6eaaeaacqGHsislcaaIXaaaaOWaaeWa aeaacaaIWaGaaGOlaiaaicdacaaI1aaacaGLOaGaayzkaaaaaa@40DB@ t = F N 1 ( 0.25 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbeqabeWacmGabiqabeqabmqabeabbaGcbaGaamiDaiaai2 dacaWGgbWaa0baaSqaaiaad6eaaeaacqGHsislcaaIXaaaaOWaaeWa aeaacaaIWaGaaGOlaiaaikdacaaI1aaacaGLOaGaayzkaaaaaa@40DD@ t = F N 1 ( 0.50 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbeqabeWacmGabiqabeqabmqabeabbaGcbaGaamiDaiaai2 dacaWGgbWaa0baaSqaaiaad6eaaeaacqGHsislcaaIXaaaaOWaaeWa aeaacaaIWaGaaGOlaiaaiwdacaaIWaaacaGLOaGaayzkaaaaaa@40DB@ t = F N 1 ( 0.75 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbeqabeWacmGabiqabeqabmqabeabbaGcbaGaamiDaiaai2 dacaWGgbWaa0baaSqaaiaad6eaaeaacqGHsislcaaIXaaaaOWaaeWa aeaacaaIWaGaaGOlaiaaiEdacaaI1aaacaGLOaGaayzkaaaaaa@40E2@ t = F N 1 ( 0.95 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbeqabeWacmGabiqabeqabmqabeabbaGcbaGaamiDaiaai2 dacaWGgbWaa0baaSqaaiaad6eaaeaacqGHsislcaaIXaaaaOWaaeWa aeaacaaIWaGaaGOlaiaaiMdacaaI1aaacaGLOaGaayzkaaaaaa@40E4@
BIAS RMSE BIAS RMSE RBIAS RMSE BIAS RMSE BIAS RMSE
MU284 population with Y = ln R M T 85 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamywaiaai2 daciGGSbGaaiOBaiaadkfacaWGnbGaamivaiaaiIdacaaI1aaaaa@3EAC@ and X = ln P 85 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiwaiaai2 daciGGSbGaaiOBaiaadcfacaaI4aGaaGynaaaa@3CFE@
F ^ s ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaja WaaSbaaSqaaiaadohaaeqaaOWaaeWaaeaacaWG0baacaGLOaGaayzk aaaaaa@3BAB@ 133 421 339 625 180 529 -265 490 -187 439
F ^ l ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaja WaaSbaaSqaaiaadYgaaeqaaOWaaeWaaeaacaWG0baacaGLOaGaayzk aaaaaa@3BA4@ 52 380 67 588 45 555 -63 469 -87 370
F ^ s * ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaja Waa0baaSqaaiaadohaaeaacaaIQaaaaOWaaeWaaeaacaWG0baacaGL OaGaayzkaaaaaa@3C60@ 8 81 -154 203 90 130 62 123 6 54
F ^ l * ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaja Waa0baaSqaaiaadYgaaeaacaaIQaaaaOWaaeWaaeaacaWG0baacaGL OaGaayzkaaaaaa@3C59@ 28 66 -170 212 69 112 57 109 2 50
F ˜ s ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaia WaaSbaaSqaaiaadohaaeqaaOWaaeWaaeaacaWG0baacaGLOaGaayzk aaaaaa@3BAA@ -28 300 -24 497 8 483 -48 421 -38 319
F ˜ l ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaia WaaSbaaSqaaiaadYgaaeqaaOWaaeWaaeaacaWG0baacaGLOaGaayzk aaaaaa@3BA3@ -28 326 -96 569 -52 544 3 466 1 319
F ˜ s * ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaia Waa0baaSqaaiaadohaaeaacaaIQaaaaOWaaeWaaeaacaWG0baacaGL OaGaayzkaaaaaa@3C5F@ 26 177 -11 302 0 244 1 308 -18 102
F ˜ l * ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaia Waa0baaSqaaiaadYgaaeaacaaIQaaaaOWaaeWaaeaacaWG0baacaGL OaGaayzkaaaaaa@3C58@ 29 179 -10 302 -2 243 -1 308 -21 104
F ˜ π ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaia WaaSbaaSqaaiabec8aWbqabaGcdaqadaqaaiaadshaaiaawIcacaGL Paaaaaa@3C6F@ 22 388 -10 771 9 864 5 731 -43 394
  MU284 population with Y = ln R M T 85 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamywaiaai2 daciGGSbGaaiOBaiaadkfacaWGnbGaamivaiaaiIdacaaI1aaaaa@3EAC@ and X = ln R E V 84 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiwaiaai2 daciGGSbGaaiOBaiaadkfacaWGfbGaamOvaiaaiIdacaaI0aaaaa@3EA4@
F ^ s ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaja WaaSbaaSqaaiaadohaaeqaaOWaaeWaaeaacaWG0baacaGLOaGaayzk aaaaaa@3BAB@ 143 449 303 643 138 554 -217 543 -166 446
F ^ l ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaja WaaSbaaSqaaiaadYgaaeqaaOWaaeWaaeaacaWG0baacaGLOaGaayzk aaaaaa@3BA4@ 62 395 62 611 36 582 -49 519 -71 376
F ^ s * ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaja Waa0baaSqaaiaadohaaeaacaaIQaaaaOWaaeWaaeaacaWG0baacaGL OaGaayzkaaaaaa@3C60@ -11 204 -32 300 -101 328 42 285 31 155
F ^ l * ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaja Waa0baaSqaaiaadYgaaeaacaaIQaaaaOWaaeWaaeaacaWG0baacaGL OaGaayzkaaaaaa@3C59@ 36 183 -40 288 -149 345 6 261 34 122
F ˜ s ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaia WaaSbaaSqaaiaadohaaeqaaOWaaeWaaeaacaWG0baacaGLOaGaayzk aaaaaa@3BAA@ 5 340 -22 548 4 557 -30 498 -23 332
F ˜ l ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaia WaaSbaaSqaaiaadYgaaeqaaOWaaeWaaeaacaWG0baacaGLOaGaayzk aaaaaa@3BA3@ -2 349 -78 599 -36 588 10 522 8 331
F ˜ s * ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaia Waa0baaSqaaiaadohaaeaacaaIQaaaaOWaaeWaaeaacaWG0baacaGL OaGaayzkaaaaaa@3C5F@ 24 303 7 446 -6 494 2 439 -13 209
F ˜ l * ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaia Waa0baaSqaaiaadYgaaeaacaaIQaaaaOWaaeWaaeaacaWG0baacaGL OaGaayzkaaaaaa@3C58@ 29 304 4 443 -6 495 -1 432 -18 192
F ˜ π ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaia WaaSbaaSqaaiabec8aWbqabaGcdaqadaqaaiaadshaaiaawIcacaGL Paaaaaa@3C6F@ 34 395 1 766 16 880 9 744 -37 398
Table 5.4
Real populations (population size N = 284 ) . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbeqabeWacmGabiqabeqabmqabeabbaGcbaWaaeGaaeaaca WGobGaaGypaiaaikdacaaI4aGaaGinaaGaayzkaaGaaiOlaaaa@3A48@ BIAS and RMSE of distribution function estimators under Poisson sampling with inclusion probabilities proportional to the absolute value of the residuals of the linear regression of the population y i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbeqabeWacmGabiqabeqabmqabeabbaGcbaGaamyEamaaBa aaleaacaWGPbaabeaakiabgkHiTaaa@3807@ values on the population x i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbeqabeWacmGabiqabeqabmqabeabbaGcbaGaamiEamaaBa aaleaacaWGPbaabeaakiabgkHiTaaa@3806@ values. Expected size n * = 30 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbeqabeWacmGabiqabeqabmqabeabbaGcbaGaamOBamaaCa aaleqabaGaaGOkaaaakiaai2dacaaIZaGaaGimaaaa@3914@
Table summary
This table displays the results of Real populations (population size XXXX BIAS and RMSE of distribution function estimators under Poisson sampling with inclusion probabilities proportional to the absolute value of the residuals of the population XXXX values on the population XXXX values. Expected size XXXX. The information is grouped by (appearing as row headers), XXXX, BIAS , RMSE , RMSE and RBIAS , calculated using MU284 population with XXXX and XXXX units of measure (appearing as column headers).
  t = F N 1 ( 0.05 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbeqabeWacmGabiqabeqabmqabeabbaGcbaGaamiDaiaai2 dacaWGgbWaa0baaSqaaiaad6eaaeaacqGHsislcaaIXaaaaOWaaeWa aeaacaaIWaGaaGOlaiaaicdacaaI1aaacaGLOaGaayzkaaaaaa@40DB@ t = F N 1 ( 0.25 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbeqabeWacmGabiqabeqabmqabeabbaGcbaGaamiDaiaai2 dacaWGgbWaa0baaSqaaiaad6eaaeaacqGHsislcaaIXaaaaOWaaeWa aeaacaaIWaGaaGOlaiaaikdacaaI1aaacaGLOaGaayzkaaaaaa@40DD@ t = F N 1 ( 0.50 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbeqabeWacmGabiqabeqabmqabeabbaGcbaGaamiDaiaai2 dacaWGgbWaa0baaSqaaiaad6eaaeaacqGHsislcaaIXaaaaOWaaeWa aeaacaaIWaGaaGOlaiaaiwdacaaIWaaacaGLOaGaayzkaaaaaa@40DB@ t = F N 1 ( 0.75 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbeqabeWacmGabiqabeqabmqabeabbaGcbaGaamiDaiaai2 dacaWGgbWaa0baaSqaaiaad6eaaeaacqGHsislcaaIXaaaaOWaaeWa aeaacaaIWaGaaGOlaiaaiEdacaaI1aaacaGLOaGaayzkaaaaaa@40E2@ t = F N 1 ( 0.95 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbeqabeWacmGabiqabeqabmqabeabbaGcbaGaamiDaiaai2 dacaWGgbWaa0baaSqaaiaad6eaaeaacqGHsislcaaIXaaaaOWaaeWa aeaacaaIWaGaaGOlaiaaiMdacaaI1aaacaGLOaGaayzkaaaaaa@40E4@
BIAS RMSE BIAS RMSE RBIAS RMSE BIAS RMSE BIAS RMSE
MU284 population with Y = ln R M T 85 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamywaiaai2 daciGGSbGaaiOBaiaadkfacaWGnbGaamivaiaaiIdacaaI1aaaaa@3EAC@ and X = ln P 85 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiwaiaai2 daciGGSbGaaiOBaiaadcfacaaI4aGaaGynaaaa@3CFE@
F ^ s ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaja WaaSbaaSqaaiaadohaaeqaaOWaaeWaaeaacaWG0baacaGLOaGaayzk aaaaaa@3BAB@ 204 420 485 668 239 519 -412 626 -90 317
F ^ l ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaja WaaSbaaSqaaiaadYgaaeqaaOWaaeWaaeaacaWG0baacaGLOaGaayzk aaaaaa@3BA4@ 180 424 417 684 319 614 -239 548 -148 348
F ^ s * ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaja Waa0baaSqaaiaadohaaeaacaaIQaaaaOWaaeWaaeaacaWG0baacaGL OaGaayzkaaaaaa@3C60@ -41 97 -118 199 132 178 40 140 -71 104
F ^ l * ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaja Waa0baaSqaaiaadYgaaeaacaaIQaaaaOWaaeWaaeaacaWG0baacaGL OaGaayzkaaaaaa@3C59@ 11 70 -147 211 63 128 -25 122 -85 106
F ˜ s ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaia WaaSbaaSqaaiaadohaaeqaaOWaaeWaaeaacaWG0baacaGLOaGaayzk aaaaaa@3BAA@ 24 360 30 649 0 675 -68 614 58 368
F ˜ l ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaia WaaSbaaSqaaiaadYgaaeqaaOWaaeWaaeaacaWG0baacaGLOaGaayzk aaaaaa@3BA3@ 9 390 -63 737 -64 774 -7 682 75 414
F ˜ s * ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaia Waa0baaSqaaiaadohaaeaacaaIQaaaaOWaaeWaaeaacaWG0baacaGL OaGaayzkaaaaaa@3C5F@ 16 184 -14 307 36 283 16 323 -11 103
F ˜ l * ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaia Waa0baaSqaaiaadYgaaeaacaaIQaaaaOWaaeWaaeaacaWG0baacaGL OaGaayzkaaaaaa@3C58@ 25 187 -15 312 30 286 14 328 -11 112
F ˜ π ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaia WaaSbaaSqaaiabec8aWbqabaGcdaqadaqaaiaadshaaiaawIcacaGL Paaaaaa@3C6F@ 40 445 73 1,983 12 2,498 -43 3,094 -49 3,341
  MU284 population with Y = ln R M T 85 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamywaiaai2 daciGGSbGaaiOBaiaadkfacaWGnbGaamivaiaaiIdacaaI1aaaaa@3EAC@ and X = ln R E V 84 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiwaiaai2 daciGGSbGaaiOBaiaadkfacaWGfbGaamOvaiaaiIdacaaI0aaaaa@3EA4@
F ^ s ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaja WaaSbaaSqaaiaadohaaeqaaOWaaeWaaeaacaWG0baacaGLOaGaayzk aaaaaa@3BAB@ 349 660 1,185 1,373 890 1,059 458 654 -32 270
F ^ l ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaja WaaSbaaSqaaiaadYgaaeqaaOWaaeWaaeaacaWG0baacaGLOaGaayzk aaaaaa@3BA4@ 287 601 1,003 1,236 771 989 484 695 42 263
F ^ s * ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaja Waa0baaSqaaiaadohaaeaacaaIQaaaaOWaaeWaaeaacaWG0baacaGL OaGaayzkaaaaaa@3C60@ 317 453 739 866 761 879 624 701 159 207
F ^ l * ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaja Waa0baaSqaaiaadYgaaeaacaaIQaaaaOWaaeWaaeaacaWG0baacaGL OaGaayzkaaaaaa@3C59@ 364 471 720 842 718 824 572 647 96 158
F ˜ s ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaia WaaSbaaSqaaiaadohaaeqaaOWaaeWaaeaacaWG0baacaGLOaGaayzk aaaaaa@3BAA@ 35 488 82 818 -31 772 7 634 -8 326
F ˜ l ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaia WaaSbaaSqaaiaadYgaaeqaaOWaaeWaaeaacaWG0baacaGLOaGaayzk aaaaaa@3BA3@ 22 500 3 878 -98 852 40 704 27 354
F ˜ s * ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaia Waa0baaSqaaiaadohaaeaacaaIQaaaaOWaaeWaaeaacaWG0baacaGL OaGaayzkaaaaaa@3C5F@ 37 317 32 498 -13 513 32 412 7 157
F ˜ l * ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaia Waa0baaSqaaiaadYgaaeaacaaIQaaaaOWaaeWaaeaacaWG0baacaGL OaGaayzkaaaaaa@3C58@ 51 313 30 498 -30 518 12 411 -10 149
F ˜ π ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaia WaaSbaaSqaaiabec8aWbqabaGcdaqadaqaaiaadshaaiaawIcacaGL Paaaaaa@3C6F@ 32 671 19 1,658 -172 2,354 -173 2,787 -191 2,935

Consider finally the simulation results referring to the variance estimators of Section 4. Tables 5.5 to 5.8 report the relative bias (RBIAS) and the relative root mean square error (RRMSE) for each of them. For example, based on the variance estimates V ˜ b ( F ˜ ( t ) ) , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOvayaaia WaaSbaaSqaaiaadkgaaeqaaOWaaeWaaeaaceWGgbGbaGaadaqadaqa aiaadshaaiaawIcacaGLPaaaaiaawIcacaGLPaaacaGGSaaaaa@3C9B@ b = 1,2, , B , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOyaiaai2 dacaaIXaGaaGilaiaaikdacaaISaGaeSOjGSKaaGilaiaadkeacaGG Saaaaa@3CDF@ obtained from the estimator V ˜ ( F ˜ ( t ) ) , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOvayaaia WaaeWaaeaaceWGgbGbaGaadaqadaqaaiaadshaaiaawIcacaGLPaaa aiaawIcacaGLPaaacaGGSaaaaa@3B7E@

RBIAS  := 1 B b = 1 B V ˜ b ( F ˜ ( t ) ) V B ( F ˜ ( t ) ) V B ( F ˜ ( t ) ) × 10,000 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeOuaiaabk eacaqGjbGaaeyqaiaabofacaaI6aGaaGypamaalaaabaGaaGymaaqa aiaadkeaaaWaaabCaeqaleaacaWGIbGaaGypaiaaigdaaeaacaWGcb aaniabggHiLdGcdaWcaaqaaiqadAfagaacamaaBaaaleaacaWGIbaa beaakmaabmaabaGabmOrayaaiaWaaeWaaeaacaWG0baacaGLOaGaay zkaaaacaGLOaGaayzkaaGaeyOeI0IaamOvamaaBaaaleaacaWGcbaa beaakmaabmaabaGabmOrayaaiaWaaeWaaeaacaWG0baacaGLOaGaay zkaaaacaGLOaGaayzkaaaabaGaamOvamaaBaaaleaacaWGcbaabeaa kmaabmaabaGabmOrayaaiaWaaeWaaeaacaWG0baacaGLOaGaayzkaa aacaGLOaGaayzkaaaaaiabgEna0kaabgdacaqGWaGaaeilaiaabcda caqGWaGaaeimaaaa@5D41@

and

RRMSE  := 1 B b = 1 B ( V ˜ b ( F ˜ ( t ) ) V B ( F ˜ ( t ) ) ) 2 V B ( F ˜ ( t ) ) × 10,000 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeOuaiaabk facaqGnbGaae4uaiaabweacaaI6aGaaGypamaalaaabaWaaOaaaeaa daWcaaqaaiaaigdaaeaacaWGcbaaamaaqahabeWcbaGaamOyaiaai2 dacaaIXaaabaGaamOqaaqdcqGHris5aOWaaeWaaeaaceWGwbGbaGaa daWgaaWcbaGaamOyaaqabaGcdaqadaqaaiqadAeagaacamaabmaaba GaamiDaaGaayjkaiaawMcaaaGaayjkaiaawMcaaiabgkHiTiaadAfa daWgaaWcbaGaamOqaaqabaGcdaqadaqaaiqadAeagaacamaabmaaba GaamiDaaGaayjkaiaawMcaaaGaayjkaiaawMcaaaGaayjkaiaawMca amaaCaaaleqabaGaaGOmaaaaaeqaaaGcbaGaamOvamaaBaaaleaaca WGcbaabeaakmaabmaabaGabmOrayaaiaWaaeWaaeaacaWG0baacaGL OaGaayzkaaaacaGLOaGaayzkaaaaaiabgEna0kaabgdacaqGWaGaae ilaiaabcdacaqGWaGaaeimaaaa@5FE5@

where

V B ( F ˜ ( t ) ) := 1 B b = 1 B ( F ˜ b ( t ) F N ( t ) ) 2 . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOvamaaBa aaleaacaWGcbaabeaakmaabmaabaGabmOrayaaiaWaaeWaaeaacaWG 0baacaGLOaGaayzkaaaacaGLOaGaayzkaaGaaGOoaiaai2dadaWcaa qaaiaaigdaaeaacaWGcbaaamaaqahabeWcbaGaamOyaiaai2dacaaI XaaabaGaamOqaaqdcqGHris5aOWaaeWaaeaaceWGgbGbaGaadaWgaa WcbaGaamOyaaqabaGcdaqadaqaaiaadshaaiaawIcacaGLPaaacqGH sislcaWGgbWaaSbaaSqaaiaad6eaaeqaaOWaaeWaaeaacaWG0baaca GLOaGaayzkaaaacaGLOaGaayzkaaWaaWbaaSqabeaacaaIYaaaaOGa aGOlaaaa@5145@

As a benchmark, we report also the RBIAS and RRMSE of the estimator

V ˜ ( F ˜ π ( t ) ) := 1 N 2 i , j s π i , j π i π j π i , j π i π j I ( y i t ) I ( y j t ) . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOvayaaia WaaeWaaeaaceWGgbGbaGaadaWgaaWcbaGaeqiWdahabeaakmaabmaa baGaamiDaaGaayjkaiaawMcaaaGaayjkaiaawMcaaiaaiQdacaaI9a WaaSaaaeaacaaIXaaabaGaamOtamaaCaaaleqabaGaaGOmaaaaaaGc daaeqbqabSqaaiaadMgacaaISaGaamOAaiabgIGiolaadohaaeqani abggHiLdGcdaWcaaqaaiabec8aWnaaBaaaleaacaWGPbGaaGilaiaa dQgaaeqaaOGaeyOeI0IaeqiWda3aaSbaaSqaaiaadMgaaeqaaOGaeq iWda3aaSbaaSqaaiaadQgaaeqaaaGcbaGaeqiWda3aaSbaaSqaaiaa dMgacaaISaGaamOAaaqabaGccqaHapaCdaWgaaWcbaGaamyAaaqaba GccqaHapaCdaWgaaWcbaGaamOAaaqabaaaaOGaamysamaabmaabaGa amyEamaaBaaaleaacaWGPbaabeaakiabgsMiJkaadshaaiaawIcaca GLPaaacaWGjbWaaeWaaeaacaWG5bWaaSbaaSqaaiaadQgaaeqaaOGa eyizImQaamiDaaGaayjkaiaawMcaaiaai6caaaa@6CAF@

for the variance of the Horvitz-Thompson estimator.

Table 5.5
Artificial populations (population size N = 1,000 ) . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbeqabeWacmGabiqabeqabmqabeabbaGcbaWaaeGaaeaaca WGobGaaGypaiaabgdacaqGSaGaaeimaiaabcdacaqGWaaacaGLPaaa caGGUaaaaa@3B88@ RBIAS and RRMSE of variance estimators under simple random without replacement sampling. Sample size n = 100 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbeqabeWacmGabiqabeqabmqabeabbaGcbaGaamOBaiaai2 dacaaIXaGaaGimaiaaicdaaaa@38E1@
Table summary
This table displays the results of Artificial populations (population size XXXX RBIAS and RRMSE of variance estimators under simple random without replacement sampling. Sample size XXXX. The information is grouped by (appearing as row headers), XXXX, RBIAS , RRMSE and RRMSE, calculated using XXXX with XXXX i.i.d. central Student XXXX with XXXX and XXXX with XXXX i.i.d. Student XXXX with XXXX units of measure (appearing as column headers).
  t = F N 1 ( 0.05 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbeqabeWacmGabiqabeqabmqabeabbaGcbaGaamiDaiaai2 dacaWGgbWaa0baaSqaaiaad6eaaeaacqGHsislcaaIXaaaaOWaaeWa aeaacaaIWaGaaGOlaiaaicdacaaI1aaacaGLOaGaayzkaaaaaa@40DB@ t = F N 1 ( 0.25 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbeqabeWacmGabiqabeqabmqabeabbaGcbaGaamiDaiaai2 dacaWGgbWaa0baaSqaaiaad6eaaeaacqGHsislcaaIXaaaaOWaaeWa aeaacaaIWaGaaGOlaiaaikdacaaI1aaacaGLOaGaayzkaaaaaa@40DD@ t = F N 1 ( 0.50 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbeqabeWacmGabiqabeqabmqabeabbaGcbaGaamiDaiaai2 dacaWGgbWaa0baaSqaaiaad6eaaeaacqGHsislcaaIXaaaaOWaaeWa aeaacaaIWaGaaGOlaiaaiwdacaaIWaaacaGLOaGaayzkaaaaaa@40DB@ t = F N 1 ( 0.75 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbeqabeWacmGabiqabeqabmqabeabbaGcbaGaamiDaiaai2 dacaWGgbWaa0baaSqaaiaad6eaaeaacqGHsislcaaIXaaaaOWaaeWa aeaacaaIWaGaaGOlaiaaiEdacaaI1aaacaGLOaGaayzkaaaaaa@40E2@ t = F N 1 ( 0.95 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbeqabeWacmGabiqabeqabmqabeabbaGcbaGaamiDaiaai2 dacaWGgbWaa0baaSqaaiaad6eaaeaacqGHsislcaaIXaaaaOWaaeWa aeaacaaIWaGaaGOlaiaaiMdacaaI1aaacaGLOaGaayzkaaaaaa@40E4@
RBIAS RRMSE RBIAS RRMSE RBIAS RRMSE RBIAS RRMSE RBIAS RRMSE
y i = ε i , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEamaaBa aaleaacaWGPbaabeaakiaai2dacqaH1oqzdaWgaaWcbaGaamyAaaqa baGccaGGSaaaaa@3D84@ with ε i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyTdu2aaS baaSqaaiaadMgaaeqaaOGaeSipIOdaaa@3B14@ i.i.d. central Student t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiDaaaa@3819@ with ν = 5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyVd4MaaG ypaiaaiwdaaaa@3A5E@
V ˜ ( F ˜ s ( t ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOvayaaia WaaeWabeaaceWGgbGbaGaadaWgaaWcbaGaam4CaaqabaGcdaqadaqa aiaadshaaiaawIcacaGLPaaaaiaawIcacaGLPaaaaaa@3E1E@ -1,092 32,442 -1,249 3,895 -1,714 3,077 -1,536 3,828 -824 34,601
V ˜ ( F ˜ l ( t ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOvayaaia WaaeWabeaaceWGgbGbaGaadaWgaaWcbaGaamiBaaqabaGcdaqadaqa aiaadshaaiaawIcacaGLPaaaaiaawIcacaGLPaaaaaa@3E17@ -576 31,726 -603 3,838 -1,122 3,374 -951 3,758 -441 33,055
V ˜ ( F ˜ s * ( t ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOvayaaia WaaeWabeaaceWGgbGbaGaadaqhaaWcbaGaam4CaaqaaiaaiQcaaaGc daqadaqaaiaadshaaiaawIcacaGLPaaaaiaawIcacaGLPaaaaaa@3ED3@ -1,091 32,579 -1,292 3,914 -1,708 3,085 -1,640 3,828 -802 34,809
V ˜ ( F ˜ l * ( t ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOvayaaia WaaeWabeaaceWGgbGbaGaadaqhaaWcbaGaamiBaaqaaiaaiQcaaaGc daqadaqaaiaadshaaiaawIcacaGLPaaaaiaawIcacaGLPaaaaaa@3ECC@ -556 31,881 -622 3,857 -1,148 3,361 -1,025 3,749 -425 33,184
V ˜ ( F ˜ π ( t ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOvayaaia WaaeWabeaaceWGgbGbaGaadaWgaaWcbaGaeqiWdahabeaakmaabmaa baGaamiDaaGaayjkaiaawMcaaaGaayjkaiaawMcaaaaa@3EE3@ 42 30,952 57 3,928 -592 3,776 -287 3,825 551 33,462
  y i = ε i , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEamaaBa aaleaacaWGPbaabeaakiaai2dacqaH1oqzdaWgaaWcbaGaamyAaaqa baGccaGGSaaaaa@3D84@ with ε i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyTdu2aaS baaSqaaiaadMgaaeqaaOGaeSipIOdaaa@3B14@ indep. noncentral Student t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiDaaaa@3819@ with ν = 5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyVd4MaaG ypaiaaiwdaaaa@3A5E@ and μ = 15 x i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiVd0MaaG ypaiaaigdacaaI1aGaamiEamaaBaaaleaacaWGPbaabeaaaaa@3D2E@
V ˜ ( F ˜ s ( t ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOvayaaia WaaeWabeaaceWGgbGbaGaadaWgaaWcbaGaam4CaaqabaGcdaqadaqa aiaadshaaiaawIcacaGLPaaaaiaawIcacaGLPaaaaaa@3E1E@ -1,900 29,622 50 4,707 -917 3,557 -998 3,695 -1,480 29,417
V ˜ ( F ˜ l ( t ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOvayaaia WaaeWabeaaceWGgbGbaGaadaWgaaWcbaGaamiBaaqabaGcdaqadaqa aiaadshaaiaawIcacaGLPaaaaiaawIcacaGLPaaaaaa@3E17@ -1,359 29,623 535 4,572 -395 3,881 -527 3,736 -1,277 28,267
V ˜ ( F ˜ s * ( t ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOvayaaia WaaeWabeaaceWGgbGbaGaadaqhaaWcbaGaam4CaaqaaiaaiQcaaaGc daqadaqaaiaadshaaiaawIcacaGLPaaaaiaawIcacaGLPaaaaaa@3ED3@ -1,832 30,119 -101 4,710 -991 3,530 -1,077 3,704 -1,398 29,927
V ˜ ( F ˜ l * ( t ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOvayaaia WaaeWabeaaceWGgbGbaGaadaqhaaWcbaGaamiBaaqaaiaaiQcaaaGc daqadaqaaiaadshaaiaawIcacaGLPaaaaiaawIcacaGLPaaaaaa@3ECC@ -1,362 29,713 465 4,559 -420 3,865 -591 3,718 -1,236 28,489
V ˜ ( F ˜ π ( t ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOvayaaia WaaeWabeaaceWGgbGbaGaadaWgaaWcbaGaeqiWdahabeaakmaabmaa baGaamiDaaGaayjkaiaawMcaaaGaayjkaiaawMcaaaaa@3EE3@ -351 29,132 1,096 4,215 -78 4,074 574 4,067 -638 29,507
  y i = 10 x i + ε i , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEamaaBa aaleaacaWGPbaabeaakiaai2dacaaIXaGaaGimaiaadIhadaWgaaWc baGaamyAaaqabaGccqGHRaWkcqaH1oqzdaWgaaWcbaGaamyAaaqaba GccaGGSaaaaa@41FC@ with ε i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyTdu2aaS baaSqaaiaadMgaaeqaaOGaeSipIOdaaa@3B14@ i.i.d. Student t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiDaaaa@3819@ with ν = 5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyVd4MaaG ypaiaaiwdaaaa@3A5E@
V ˜ ( F ˜ s ( t ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOvayaaia WaaeWabeaaceWGgbGbaGaadaWgaaWcbaGaam4CaaqabaGcdaqadaqa aiaadshaaiaawIcacaGLPaaaaiaawIcacaGLPaaaaaa@3E1E@ -2,170 11,624 -1,027 2,480 -816 3,274 -1,424 2,583 -1,946 8,681
V ˜ ( F ˜ l ( t ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOvayaaia WaaeWabeaaceWGgbGbaGaadaWgaaWcbaGaamiBaaqabaGcdaqadaqa aiaadshaaiaawIcacaGLPaaaaiaawIcacaGLPaaaaaa@3E17@ -1,534 11,605 -529 2,632 -148 2,975 -859 2,590 -1,151 9,015
V ˜ ( F ˜ s * ( t ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOvayaaia WaaeWabeaaceWGgbGbaGaadaqhaaWcbaGaam4CaaqaaiaaiQcaaaGc daqadaqaaiaadshaaiaawIcacaGLPaaaaiaawIcacaGLPaaaaaa@3ED3@ -1,765 12,107 -1,108 2,529 -714 3,366 -1,318 2,660 -1,905 8,658
V ˜ ( F ˜ l * ( t ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOvayaaia WaaeWabeaaceWGgbGbaGaadaqhaaWcbaGaamiBaaqaaiaaiQcaaaGc daqadaqaaiaadshaaiaawIcacaGLPaaaaiaawIcacaGLPaaaaaa@3ECC@ -1,062 11,948 -671 2,735 -212 3,291 -762 2,785 -1,048 8,590
V ˜ ( F ˜ π ( t ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOvayaaia WaaeWabeaaceWGgbGbaGaadaWgaaWcbaGaeqiWdahabeaakmaabmaa baGaamiDaaGaayjkaiaawMcaaaGaayjkaiaawMcaaaaa@3EE3@ 254 31,545 -52 3,726 136 4,152 267 3,992 35 30,264
  y i = 10 x i + ε i , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEamaaBa aaleaacaWGPbaabeaakiaai2dacaaIXaGaaGimaiaadIhadaWgaaWc baGaamyAaaqabaGccqGHRaWkcqaH1oqzdaWgaaWcbaGaamyAaaqaba GccaGGSaaaaa@41FC@ with ε i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyTdu2aaS baaSqaaiaadMgaaeqaaOGaeSipIOdaaa@3B14@ indep. noncentral Student t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiDaaaa@3819@ with ν = 5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyVd4MaaG ypaiaaiwdaaaa@3A5E@ and μ = 15 x i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiVd0MaaG ypaiaaigdacaaI1aGaamiEamaaBaaaleaacaWGPbaabeaaaaa@3D2E@
V ˜ ( F ˜ s ( t ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOvayaaia WaaeWabeaaceWGgbGbaGaadaWgaaWcbaGaam4CaaqabaGcdaqadaqa aiaadshaaiaawIcacaGLPaaaaiaawIcacaGLPaaaaaa@3E1E@ -1,642 25,809 -855 3,541 -1,076 3,038 -1,081 3,030 -1,361 21,157
V ˜ ( F ˜ l ( t ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOvayaaia WaaeWabeaaceWGgbGbaGaadaWgaaWcbaGaamiBaaqabaGcdaqadaqa aiaadshaaiaawIcacaGLPaaaaiaawIcacaGLPaaaaaa@3E17@ -950 25,692 -323 3,509 -597 3,312 -617 3,164 -1,124 20,231
V ˜ ( F ˜ s * ( t ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOvayaaia WaaeWabeaaceWGgbGbaGaadaqhaaWcbaGaam4CaaqaaiaaiQcaaaGc daqadaqaaiaadshaaiaawIcacaGLPaaaaiaawIcacaGLPaaaaaa@3ED3@ -1,385 26,406 -997 3,505 -1,089 3,045 -1,096 3,033 -1,310 21,393
V ˜ ( F ˜ l * ( t ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOvayaaia WaaeWabeaaceWGgbGbaGaadaqhaaWcbaGaamiBaaqaaiaaiQcaaaGc daqadaqaaiaadshaaiaawIcacaGLPaaaaiaawIcacaGLPaaaaaa@3ECC@ -832 26,212 -292 3,556 -614 3,317 -716 3,154 -1,135 20,286
V ˜ ( F ˜ π ( t ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOvayaaia WaaeWabeaaceWGgbGbaGaadaWgaaWcbaGaeqiWdahabeaakmaabmaa baGaamiDaaGaayjkaiaawMcaaaGaayjkaiaawMcaaaaa@3EE3@ 105 29,621 507 3,857 209 4,244 425 3,910 -337 29,082
  y i = 10 x i 1 / 4 + ε i , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEamaaBa aaleaacaWGPbaabeaakiaai2dacaaIXaGaaGimaiaadIhadaqhaaWc baGaamyAaaqaamaalyaabaGaaGymaaqaaiaaisdaaaaaaOGaey4kaS IaeqyTdu2aaSbaaSqaaiaadMgaaeqaaOGaaiilaaaa@438C@ with ε i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyTdu2aaS baaSqaaiaadMgaaeqaaOGaeSipIOdaaa@3B14@ i.i.d. Student t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiDaaaa@3819@ with ν = 5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyVd4MaaG ypaiaaiwdaaaa@3A5E@
V ˜ ( F ˜ s ( t ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOvayaaia WaaeWabeaaceWGgbGbaGaadaWgaaWcbaGaam4CaaqabaGcdaqadaqa aiaadshaaiaawIcacaGLPaaaaiaawIcacaGLPaaaaaa@3E1E@ -2,465 30,612 -1,121 4,594 -1,512 3,183 -1,958 3,076 -863 19,720
V ˜ ( F ˜ l ( t ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOvayaaia WaaeWabeaaceWGgbGbaGaadaWgaaWcbaGaamiBaaqabaGcdaqadaqa aiaadshaaiaawIcacaGLPaaaaiaawIcacaGLPaaaaaa@3E17@ -1,780 28,103 -663 4,420 -1,092 3,319 -1,491 3,140 -439 18,985
V ˜ ( F ˜ s * ( t ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOvayaaia WaaeWabeaaceWGgbGbaGaadaqhaaWcbaGaam4CaaqaaiaaiQcaaaGc daqadaqaaiaadshaaiaawIcacaGLPaaaaiaawIcacaGLPaaaaaa@3ED3@ -2,052 33,980 -1,150 4,619 -1,537 3,217 -1,948 3,127 -954 19,637
V ˜ ( F ˜ l * ( t ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOvayaaia WaaeWabeaaceWGgbGbaGaadaqhaaWcbaGaamiBaaqaaiaaiQcaaaGc daqadaqaaiaadshaaiaawIcacaGLPaaaaiaawIcacaGLPaaaaaa@3ECC@ -1,194 33,573 -691 4,472 -1,124 3,368 -1,438 3,228 -357 19,245
V ˜ ( F ˜ π ( t ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOvayaaia WaaeWabeaaceWGgbGbaGaadaWgaaWcbaGaeqiWdahabeaakmaabmaa baGaamiDaaGaayjkaiaawMcaaaGaayjkaiaawMcaaaaa@3EE3@ -81 30,001 9 3,756 -110 3,996 -598 3,661 440 32,455
  y i = 10 x i 1 / 4 + ε i , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEamaaBa aaleaacaWGPbaabeaakiaai2dacaaIXaGaaGimaiaadIhadaqhaaWc baGaamyAaaqaamaalyaabaGaaGymaaqaaiaaisdaaaaaaOGaey4kaS IaeqyTdu2aaSbaaSqaaiaadMgaaeqaaOGaaiilaaaa@438C@ with ε i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyTdu2aaS baaSqaaiaadMgaaeqaaOGaeSipIOdaaa@3B14@ indep. noncentral Student t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiDaaaa@3819@ with ν = 5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyVd4MaaG ypaiaaiwdaaaa@3A5E@ and μ = 15 x i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiVd0MaaG ypaiaaigdacaaI1aGaamiEamaaBaaaleaacaWGPbaabeaaaaa@3D2E@
V ˜ ( F ˜ s ( t ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOvayaaia WaaeWabeaaceWGgbGbaGaadaWgaaWcbaGaam4CaaqabaGcdaqadaqa aiaadshaaiaawIcacaGLPaaaaiaawIcacaGLPaaaaaa@3E1E@ -1,873 29,437 -758 3,759 -621 3,476 -709 3,599 -1,298 27,679
V ˜ ( F ˜ l ( t ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOvayaaia WaaeWabeaaceWGgbGbaGaadaWgaaWcbaGaamiBaaqabaGcdaqadaqa aiaadshaaiaawIcacaGLPaaaaiaawIcacaGLPaaaaaa@3E17@ -1,267 28,511 -284 3,661 -131 3,758 -321 3,552 -1,075 26,790
V ˜ ( F ˜ s * ( t ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOvayaaia WaaeWabeaaceWGgbGbaGaadaqhaaWcbaGaam4CaaqaaiaaiQcaaaGc daqadaqaaiaadshaaiaawIcacaGLPaaaaiaawIcacaGLPaaaaaa@3ED3@ -1,710 30,670 -928 3,741 -628 3,510 -777 3,603 -1,245 27,972
V ˜ ( F ˜ l * ( t ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOvayaaia WaaeWabeaaceWGgbGbaGaadaqhaaWcbaGaamiBaaqaaiaaiQcaaaGc daqadaqaaiaadshaaiaawIcacaGLPaaaaiaawIcacaGLPaaaaaa@3ECC@ -939 30,486 -270 3,764 -171 3,803 -375 3,581 -1,014 26,926
V ˜ ( F ˜ π ( t ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOvayaaia WaaeWabeaaceWGgbGbaGaadaWgaaWcbaGaeqiWdahabeaakmaabmaa baGaamiDaaGaayjkaiaawMcaaaGaayjkaiaawMcaaaaa@3EE3@ 178 29,640 599 3,816 533 4,324 590 3,874 -404 28,917
Table 5.6
Artificial populations (population size N = 1,000 ) . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbeqabeWacmGabiqabeqabmqabeabbaGcbaWaaeGaaeaaca WGobGaaGypaiaabgdacaqGSaGaaeimaiaabcdacaqGWaaacaGLPaaa caGGUaaaaa@3B88@ RBIAS and RRMSE of variance estimators under Poisson sampling with sample inclusion probabilities π i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbeqabeWacmGabiqabeqabmqabeabbaGcbaGaeqiWda3aaS baaSqaaiaadMgaaeqaaaaa@37CF@ proportional to standard deviation of noncentral Student t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbeqabeWacmGabiqabeqabmqabeabbaGcbaGaamiDaaaa@35F1@ distribution with ν = 5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbeqabeWacmGabiqabeqabmqabeabbaGcbaGaeqyVd4MaaG ypaiaaiwdaaaa@3836@ d.f. and with noncentrality parameter μ = 15 x i . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbeqabeWacmGabiqabeqabmqabeabbaGcbaGaeqiVd0MaaG ypaiaaigdacaaI1aGaamiEamaaBaaaleaacaWGPbaabeaakiaac6ca aaa@3BC2@ Expected sample size n * = 100 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbeqabeWacmGabiqabeqabmqabeabbaGcbaGaamOBamaaCa aaleqabaGaaGOkaaaakiaai2dacaaIXaGaaGimaiaaicdaaaa@39CC@
Table summary
This table displays the results of Artificial populations (population size XXXX RBIAS and RRMSE of variance estimators under Poisson sampling with sample inclusion probabilities XXXX proportional to standard deviation of noncentral Student XXXX distribution with XXXX d.f. and with noncentrality parameter XXXX Expected sample size XXXX XXXX, RBIAS , RRMSE and RRMSE, calculated using XXXX with XXXX i.i.d. central Student XXXX with XXXX units of measure (appearing as column headers).
  t = F N 1 ( 0.05 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbeqabeWacmGabiqabeqabmqabeabbaGcbaGaamiDaiaai2 dacaWGgbWaa0baaSqaaiaad6eaaeaacqGHsislcaaIXaaaaOWaaeWa aeaacaaIWaGaaGOlaiaaicdacaaI1aaacaGLOaGaayzkaaaaaa@40DB@ t = F N 1 ( 0.25 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbeqabeWacmGabiqabeqabmqabeabbaGcbaGaamiDaiaai2 dacaWGgbWaa0baaSqaaiaad6eaaeaacqGHsislcaaIXaaaaOWaaeWa aeaacaaIWaGaaGOlaiaaikdacaaI1aaacaGLOaGaayzkaaaaaa@40DD@ t = F N 1 ( 0.50 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbeqabeWacmGabiqabeqabmqabeabbaGcbaGaamiDaiaai2 dacaWGgbWaa0baaSqaaiaad6eaaeaacqGHsislcaaIXaaaaOWaaeWa aeaacaaIWaGaaGOlaiaaiwdacaaIWaaacaGLOaGaayzkaaaaaa@40DB@ t = F N 1 ( 0.75 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbeqabeWacmGabiqabeqabmqabeabbaGcbaGaamiDaiaai2 dacaWGgbWaa0baaSqaaiaad6eaaeaacqGHsislcaaIXaaaaOWaaeWa aeaacaaIWaGaaGOlaiaaiEdacaaI1aaacaGLOaGaayzkaaaaaa@40E2@ t = F N 1 ( 0.95 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbeqabeWacmGabiqabeqabmqabeabbaGcbaGaamiDaiaai2 dacaWGgbWaa0baaSqaaiaad6eaaeaacqGHsislcaaIXaaaaOWaaeWa aeaacaaIWaGaaGOlaiaaiMdacaaI1aaacaGLOaGaayzkaaaaaa@40E4@
RBIAS RRMSE RBIAS RRMSE RBIAS RRMSE RBIAS RRMSE RBIAS RRMSE
y i = ε i , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEamaaBa aaleaacaWGPbaabeaakiaai2dacqaH1oqzdaWgaaWcbaGaamyAaaqa baGccaGGSaaaaa@3D84@ with ε i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyTdu2aaS baaSqaaiaadMgaaeqaaOGaeSipIOdaaa@3B14@ i.i.d. central Student t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiDaaaa@3819@ with ν = 5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyVd4MaaG ypaiaaiwdaaaa@3A5E@
V ˜ ( F ˜ s ( t ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOvayaaia WaaeWabeaaceWGgbGbaGaadaWgaaWcbaGaam4CaaqabaGcdaqadaqa aiaadshaaiaawIcacaGLPaaaaiaawIcacaGLPaaaaaa@3E1E@ -3,306 65,777 -4,248 8,032 -5,093 4,242 -6,258 4,844 -5,652 32,037
V ˜ ( F ˜ l ( t ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOvayaaia WaaeWabeaaceWGgbGbaGaadaWgaaWcbaGaamiBaaqabaGcdaqadaqa aiaadshaaiaawIcacaGLPaaaaiaawIcacaGLPaaaaaa@3E17@ -2,048 47,035 -2,656 4,705 -2,434 3,116 -3,310 3,939 -3,092 29,380
V ˜ ( F ˜ s * ( t ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOvayaaia WaaeWabeaaceWGgbGbaGaadaqhaaWcbaGaam4CaaqaaiaaiQcaaaGc daqadaqaaiaadshaaiaawIcacaGLPaaaaiaawIcacaGLPaaaaaa@3ED3@ -3,362 36,855 -2,488 4,409 -1,910 3,147 -2,869 3,910 -4,329 23,247
V ˜ ( F ˜ l * ( t ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOvayaaia WaaeWabeaaceWGgbGbaGaadaqhaaWcbaGaamiBaaqaaiaaiQcaaaGc daqadaqaaiaadshaaiaawIcacaGLPaaaaiaawIcacaGLPaaaaaa@3ECC@ -2,696 39,509 -2,076 4,450 -1,768 3,163 -2,648 3,811 -3,244 26,343
V ˜ ( F ˜ π ( t ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOvayaaia WaaeWabeaaceWGgbGbaGaadaWgaaWcbaGaeqiWdahabeaakmaabmaa baGaamiDaaGaayjkaiaawMcaaaGaayjkaiaawMcaaaaa@3EE3@ 113 129,637 259 15,120 618 6,327 193 5,429 273 6,097
  y i = ε i , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEamaaBa aaleaacaWGPbaabeaakiaai2dacqaH1oqzdaWgaaWcbaGaamyAaaqa baGccaGGSaaaaa@3D84@ with ε i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyTdu2aaS baaSqaaiaadMgaaeqaaOGaeSipIOdaaa@3B14@ indep. noncentral Student t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiDaaaa@3819@ with ν = 5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyVd4MaaG ypaiaaiwdaaaa@3A5E@ and μ = 15 x i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiVd0MaaG ypaiaaigdacaaI1aGaamiEamaaBaaaleaacaWGPbaabeaaaaa@3D2E@
V ˜ ( F ˜ s ( t ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOvayaaia WaaeWabeaaceWGgbGbaGaadaWgaaWcbaGaam4CaaqabaGcdaqadaqa aiaadshaaiaawIcacaGLPaaaaiaawIcacaGLPaaaaaa@3E1E@ -740 125,975 -2,522 14,864 -5,466 3,658 -4,896 6,691 -1,551 83,262
V ˜ ( F ˜ l ( t ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOvayaaia WaaeWabeaaceWGgbGbaGaadaWgaaWcbaGaamiBaaqabaGcdaqadaqa aiaadshaaiaawIcacaGLPaaaaiaawIcacaGLPaaaaaa@3E17@ -391 83,047 -1,503 8,946 -2,428 4,099 -2,228 5,526 -1,154 54,680
V ˜ ( F ˜ s * ( t ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOvayaaia WaaeWabeaaceWGgbGbaGaadaqhaaWcbaGaam4CaaqaaiaaiQcaaaGc daqadaqaaiaadshaaiaawIcacaGLPaaaaiaawIcacaGLPaaaaaa@3ED3@ -3,260 58,072 -2,649 7,661 -2,260 3,936 -2,795 5,011 -2,116 48,739
V ˜ ( F ˜ l * ( t ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOvayaaia WaaeWabeaaceWGgbGbaGaadaqhaaWcbaGaamiBaaqaaiaaiQcaaaGc daqadaqaaiaadshaaiaawIcacaGLPaaaaiaawIcacaGLPaaaaaa@3ECC@ -716 77,935 -2,000 7,979 -1,934 4,235 -2,279 5,243 -1,243 52,531
V ˜ ( F ˜ π ( t ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOvayaaia WaaeWabeaaceWGgbGbaGaadaWgaaWcbaGaeqiWdahabeaakmaabmaa baGaamiDaaGaayjkaiaawMcaaaGaayjkaiaawMcaaaaa@3EE3@ 666 251,134 -564 26,553 -87 7,344 -2 6,029 407 6,610
  y i = 10 x i + ε i , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEamaaBa aaleaacaWGPbaabeaakiaai2dacaaIXaGaaGimaiaadIhadaWgaaWc baGaamyAaaqabaGccqGHRaWkcqaH1oqzdaWgaaWcbaGaamyAaaqaba GccaGGSaaaaa@41FC@ with ε i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyTdu2aaS baaSqaaiaadMgaaeqaaOGaeSipIOdaaa@3B14@ i.i.d. Student t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiDaaaa@3819@ with ν = 5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyVd4MaaG ypaiaaiwdaaaa@3A5E@
V ˜ ( F ˜ s ( t ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOvayaaia WaaeWabeaaceWGgbGbaGaadaWgaaWcbaGaam4CaaqabaGcdaqadaqa aiaadshaaiaawIcacaGLPaaaaiaawIcacaGLPaaaaaa@3E1E@ -6,801 7,898 -6,470 4,281 -1,059 22,596 -398 32,401 -1,650 72,632
V ˜ ( F ˜ l ( t ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOvayaaia WaaeWabeaaceWGgbGbaGaadaWgaaWcbaGaamiBaaqabaGcdaqadaqa aiaadshaaiaawIcacaGLPaaaaiaawIcacaGLPaaaaaa@3E17@ -4,978 5,826 -2,898 4,473 -603 9,530 206 15,226 -1,157 40,466
V ˜ ( F ˜ s * ( t ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOvayaaia WaaeWabeaaceWGgbGbaGaadaqhaaWcbaGaam4CaaqaaiaaiQcaaaGc daqadaqaaiaadshaaiaawIcacaGLPaaaaiaawIcacaGLPaaaaaa@3ED3@ -4,520 6,691 -2,710 4,213 -3,245 6,723 -1,156 12,681 -2,458 32,907
V ˜ ( F ˜ l * ( t ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOvayaaia WaaeWabeaaceWGgbGbaGaadaqhaaWcbaGaamiBaaqaaiaaiQcaaaGc daqadaqaaiaadshaaiaawIcacaGLPaaaaiaawIcacaGLPaaaaaa@3ECC@ -4,226 6,206 -1,674 5,062 -978 7,874 55 12,781 -1,283 33,737
V ˜ ( F ˜ π ( t ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOvayaaia WaaeWabeaaceWGgbGbaGaadaWgaaWcbaGaeqiWdahabeaakmaabmaa baGaamiDaaGaayjkaiaawMcaaaGaayjkaiaawMcaaaaa@3EE3@ -707 47,550 118 7,214 609 4,409 743 4,628 435 4,800
  y i = 10 x i + ε i , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEamaaBa aaleaacaWGPbaabeaakiaai2dacaaIXaGaaGimaiaadIhadaWgaaWc baGaamyAaaqabaGccqGHRaWkcqaH1oqzdaWgaaWcbaGaamyAaaqaba GccaGGSaaaaa@41FC@ with ε i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyTdu2aaS baaSqaaiaadMgaaeqaaOGaeSipIOdaaa@3B14@ indep. noncentral Student t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiDaaaa@3819@ with ν = 5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyVd4MaaG ypaiaaiwdaaaa@3A5E@ and μ = 15 x i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiVd0MaaG ypaiaaigdacaaI1aGaamiEamaaBaaaleaacaWGPbaabeaaaaa@3D2E@
V ˜ ( F ˜ s ( t ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOvayaaia WaaeWabeaaceWGgbGbaGaadaWgaaWcbaGaam4CaaqabaGcdaqadaqa aiaadshaaiaawIcacaGLPaaaaiaawIcacaGLPaaaaaa@3E1E@ -7,398 8,847 -6,235 3,667 -2,493 8,171 -1,051 16,299 -1,440 71,943
V ˜ ( F ˜ l ( t ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOvayaaia WaaeWabeaaceWGgbGbaGaadaWgaaWcbaGaamiBaaqabaGcdaqadaqa aiaadshaaiaawIcacaGLPaaaaiaawIcacaGLPaaaaaa@3E17@ -4,548 9,463 -3,136 3,282 -1,187 4,246 -832 7,638 -982 45,182
V ˜ ( F ˜ s * ( t ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOvayaaia WaaeWabeaaceWGgbGbaGaadaqhaaWcbaGaam4CaaqaaiaaiQcaaaGc daqadaqaaiaadshaaiaawIcacaGLPaaaaiaawIcacaGLPaaaaaa@3ED3@ -3,902 11,727 -2,808 3,409 -2,411 3,501 -1,721 6,737 -1,671 41,389
V ˜ ( F ˜ l * ( t ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOvayaaia WaaeWabeaaceWGgbGbaGaadaqhaaWcbaGaamiBaaqaaiaaiQcaaaGc daqadaqaaiaadshaaiaawIcacaGLPaaaaiaawIcacaGLPaaaaaa@3ECC@ -3,598 10,771 -2,610 3,462 -1,284 3,988 -852 7,008 -972 43,017
V ˜ ( F ˜ π ( t ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOvayaaia WaaeWabeaaceWGgbGbaGaadaWgaaWcbaGaeqiWdahabeaakmaabmaa baGaamiDaaGaayjkaiaawMcaaaGaayjkaiaawMcaaaaa@3EE3@ 146 57,044 -42 8,708 520 4,784 214 4,686 390 5,085
  y i = 10 x i 1 / 4 + ε i , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEamaaBa aaleaacaWGPbaabeaakiaai2dacaaIXaGaaGimaiaadIhadaqhaaWc baGaamyAaaqaamaalyaabaGaaGymaaqaaiaaisdaaaaaaOGaey4kaS IaeqyTdu2aaSbaaSqaaiaadMgaaeqaaOGaaiilaaaa@438C@ with ε i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyTdu2aaS baaSqaaiaadMgaaeqaaOGaeSipIOdaaa@3B14@ i.i.d. Student t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiDaaaa@3819@ with ν = 5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyVd4MaaG ypaiaaiwdaaaa@3A5E@
V ˜ ( F ˜ s ( t ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOvayaaia WaaeWabeaaceWGgbGbaGaadaWgaaWcbaGaam4CaaqabaGcdaqadaqa aiaadshaaiaawIcacaGLPaaaaiaawIcacaGLPaaaaaa@3E1E@ -7,731 8,568 -6,597 3,484 -2,442 7,775 -903 16,067 -1,967 56,480
V ˜ ( F ˜ l ( t ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOvayaaia WaaeWabeaaceWGgbGbaGaadaWgaaWcbaGaamiBaaqabaGcdaqadaqa aiaadshaaiaawIcacaGLPaaaaiaawIcacaGLPaaaaaa@3E17@ -4,611 9,378 -2,990 3,252 -874 4,119 -347 7,420 -1,310 35,051
V ˜ ( F ˜ s * ( t ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOvayaaia WaaeWabeaaceWGgbGbaGaadaqhaaWcbaGaam4CaaqaaiaaiQcaaaGc daqadaqaaiaadshaaiaawIcacaGLPaaaaiaawIcacaGLPaaaaaa@3ED3@ -4,747 11,909 -2,679 3,298 -1,896 3,272 -2,248 5,747 -3,382 27,222
V ˜ ( F ˜ l * ( t ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOvayaaia WaaeWabeaaceWGgbGbaGaadaqhaaWcbaGaamiBaaqaaiaaiQcaaaGc daqadaqaaiaadshaaiaawIcacaGLPaaaaiaawIcacaGLPaaaaaa@3ECC@ -4,223 10,380 -2,100 3,494 -788 3,731 -550 5,975 -1,795 29,856
V ˜ ( F ˜ π ( t ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOvayaaia WaaeWabeaaceWGgbGbaGaadaWgaaWcbaGaeqiWdahabeaakmaabmaa baGaamiDaaGaayjkaiaawMcaaaGaayjkaiaawMcaaaaa@3EE3@ -428 47,038 -206 7,350 641 4,504 738 4,708 487 4,943
  y i = 10 x i 1 / 4 + ε i , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEamaaBa aaleaacaWGPbaabeaakiaai2dacaaIXaGaaGimaiaadIhadaqhaaWc baGaamyAaaqaamaalyaabaGaaGymaaqaaiaaisdaaaaaaOGaey4kaS IaeqyTdu2aaSbaaSqaaiaadMgaaeqaaOGaaiilaaaa@438C@ with ε i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyTdu2aaS baaSqaaiaadMgaaeqaaOGaeSipIOdaaa@3B14@ indep. noncentral Student t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiDaaaa@3819@ with ν = 5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyVd4MaaG ypaiaaiwdaaaa@3A5E@ and μ = 15 x i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiVd0MaaG ypaiaaigdacaaI1aGaamiEamaaBaaaleaacaWGPbaabeaaaaa@3D2E@
V ˜ ( F ˜ s ( t ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOvayaaia WaaeWabeaaceWGgbGbaGaadaWgaaWcbaGaam4CaaqabaGcdaqadaqa aiaadshaaiaawIcacaGLPaaaaiaawIcacaGLPaaaaaa@3E1E@ -4,936 40,696 -6,111 4,579 -5,549 4,035 -1,864 14,381 -1,509 84,892
V ˜ ( F ˜ l ( t ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOvayaaia WaaeWabeaaceWGgbGbaGaadaWgaaWcbaGaamiBaaqabaGcdaqadaqa aiaadshaaiaawIcacaGLPaaaaiaawIcacaGLPaaaaaa@3E17@ -3,004 29,404 -2,764 3,962 -2,436 3,606 -1,234 7,357 -1,103 53,875
V ˜ ( F ˜ s * ( t ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOvayaaia WaaeWabeaaceWGgbGbaGaadaqhaaWcbaGaam4CaaqaaiaaiQcaaaGc daqadaqaaiaadshaaiaawIcacaGLPaaaaiaawIcacaGLPaaaaaa@3ED3@ -4,328 27,704 -2,516 4,235 -2,671 3,332 -2,586 5,955 -1,939 47,601
V ˜ ( F ˜ l * ( t ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOvayaaia WaaeWabeaaceWGgbGbaGaadaqhaaWcbaGaamiBaaqaaiaaiQcaaaGc daqadaqaaiaadshaaiaawIcacaGLPaaaaiaawIcacaGLPaaaaaa@3ECC@ -3,454 28,267 -2,263 4,160 -2,329 3,574 -1,433 6,682 -1,171 50,985
V ˜ ( F ˜ π ( t ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOvayaaia WaaeWabeaaceWGgbGbaGaadaWgaaWcbaGaeqiWdahabeaakmaabmaa baGaamiDaaGaayjkaiaawMcaaaGaayjkaiaawMcaaaaa@3EE3@ 152 98,607 663 12,879 15 5,376 20 5,080 429 5,619
Table 5.7
Real populations (population size N = 284 ) . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbeqabeWacmGabiqabeqabmqabeabbaGcbaWaaeGaaeaaca WGobGaaGypaiaaikdacaaI4aGaaGinaaGaayzkaaGaaiOlaaaa@3A48@ RBIAS and RRMSE of variance estimators under simple random without replacement sampling. Sample size n = 30 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbeqabeWacmGabiqabeqabmqabeabbaGcbaGaamOBaiaai2 dacaaIZaGaaGimaaaa@3829@
Table summary
This table displays the results of Real populations (population size XXXX RBIAS and RRMSE of variance estimators under simple random without replacement sampling. Sample size XXXX XXXX, RBIAS , RRMSE and RRMSE, calculated using MU284 population with XXXX and XXXX and MU284 population with XXXX and XXXX units of measure (appearing as column headers).
  t = F N 1 ( 0.05 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbeqabeWacmGabiqabeqabmqabeabbaGcbaGaamiDaiaai2 dacaWGgbWaa0baaSqaaiaad6eaaeaacqGHsislcaaIXaaaaOWaaeWa aeaacaaIWaGaaGOlaiaaicdacaaI1aaacaGLOaGaayzkaaaaaa@40DB@ t = F N 1 ( 0.25 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbeqabeWacmGabiqabeqabmqabeabbaGcbaGaamiDaiaai2 dacaWGgbWaa0baaSqaaiaad6eaaeaacqGHsislcaaIXaaaaOWaaeWa aeaacaaIWaGaaGOlaiaaikdacaaI1aaacaGLOaGaayzkaaaaaa@40DD@ t = F N 1 ( 0.50 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbeqabeWacmGabiqabeqabmqabeabbaGcbaGaamiDaiaai2 dacaWGgbWaa0baaSqaaiaad6eaaeaacqGHsislcaaIXaaaaOWaaeWa aeaacaaIWaGaaGOlaiaaiwdacaaIWaaacaGLOaGaayzkaaaaaa@40DB@ t = F N 1 ( 0.75 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbeqabeWacmGabiqabeqabmqabeabbaGcbaGaamiDaiaai2 dacaWGgbWaa0baaSqaaiaad6eaaeaacqGHsislcaaIXaaaaOWaaeWa aeaacaaIWaGaaGOlaiaaiEdacaaI1aaacaGLOaGaayzkaaaaaa@40E2@ t = F N 1 ( 0.95 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbeqabeWacmGabiqabeqabmqabeabbaGcbaGaamiDaiaai2 dacaWGgbWaa0baaSqaaiaad6eaaeaacqGHsislcaaIXaaaaOWaaeWa aeaacaaIWaGaaGOlaiaaiMdacaaI1aaacaGLOaGaayzkaaaaaa@40E4@
RBIAS RRMSE RBIAS RRMSE RBIAS RRMSE RBIAS RRMSE RBIAS RRMSE
MU284 population with Y = ln R M T 85 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamywaiaai2 daciGGSbGaaiOBaiaadkfacaWGnbGaamivaiaaiIdacaaI1aaaaa@3EAC@ and X = ln P 85 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiwaiaai2 daciGGSbGaaiOBaiaadcfacaaI4aGaaGynaaaa@3CFE@
V ˜ ( F ˜ s ( t ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOvayaaia WaaeWabeaaceWGgbGbaGaadaWgaaWcbaGaam4CaaqabaGcdaqadaqa aiaadshaaiaawIcacaGLPaaaaiaawIcacaGLPaaaaaa@3E1E@ -2,853 16,809 -1,700 3,037 -1,554 2,984 -1,100 4,633 -5,503 16,257
V ˜ ( F ˜ l ( t ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOvayaaia WaaeWabeaaceWGgbGbaGaadaWgaaWcbaGaamiBaaqabaGcdaqadaqa aiaadshaaiaawIcacaGLPaaaaiaawIcacaGLPaaaaaa@3E17@ -1,110 16,374 -1,827 2,760 -1,683 2,847 -927 4,387 -3,016 18,685
V ˜ ( F ˜ s * ( t ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOvayaaia WaaeWabeaaceWGgbGbaGaadaqhaaWcbaGaam4CaaqaaiaaiQcaaaGc daqadaqaaiaadshaaiaawIcacaGLPaaaaiaawIcacaGLPaaaaaa@3ED3@ -1,043 19,081 -91 7,728 -448 9,120 -484 7,715 -1,877 65,298
V ˜ ( F ˜ l * ( t ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOvayaaia WaaeWabeaaceWGgbGbaGaadaqhaaWcbaGaamiBaaqaaiaaiQcaaaGc daqadaqaaiaadshaaiaawIcacaGLPaaaaiaawIcacaGLPaaaaaa@3ECC@ -424 18,971 104 7,819 -382 9,110 -301 7,799 -1,058 62,968
V ˜ ( F ˜ π ( t ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOvayaaia WaaeWabeaaceWGgbGbaGaadaWgaaWcbaGaeqiWdahabeaakmaabmaa baGaamiDaaGaayjkaiaawMcaaaGaayjkaiaawMcaaaaa@3EE3@ -186 29,720 -603 3,901 31 3,971 500 4,383 -74 28,418
  MU284 population with Y = ln R M T 85 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamywaiaai2 daciGGSbGaaiOBaiaadkfacaWGnbGaamivaiaaiIdacaaI1aaaaa@3EAC@ and X = ln R E V 84 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiwaiaai2 daciGGSbGaaiOBaiaadkfacaWGfbGaamOvaiaaiIdacaaI0aaaaa@3EA4@
V ˜ ( F ˜ s ( t ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOvayaaia WaaeWabeaaceWGgbGbaGaadaWgaaWcbaGaam4CaaqabaGcdaqadaqa aiaadshaaiaawIcacaGLPaaaaiaawIcacaGLPaaaaaa@3E1E@ -2,283 16,303 -1,450 3,538 -945 3,526 -1,071 4,300 -4,832 19,401
V ˜ ( F ˜ l ( t ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOvayaaia WaaeWabeaaceWGgbGbaGaadaWgaaWcbaGaamiBaaqabaGcdaqadaqa aiaadshaaiaawIcacaGLPaaaaiaawIcacaGLPaaaaaa@3E17@ -1,095 16,755 -1,427 3,181 -938 3,390 -780 4,051 -2,753 20,551
V ˜ ( F ˜ s * ( t ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOvayaaia WaaeWabeaaceWGgbGbaGaadaqhaaWcbaGaam4CaaqaaiaaiQcaaaGc daqadaqaaiaadshaaiaawIcacaGLPaaaaiaawIcacaGLPaaaaaa@3ED3@ -1,737 14,642 -298 5,648 -546 5,282 -736 5,679 -3,564 38,344
V ˜ ( F ˜ l * ( t ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOvayaaia WaaeWabeaaceWGgbGbaGaadaqhaaWcbaGaamiBaaqaaiaaiQcaaaGc daqadaqaaiaadshaaiaawIcacaGLPaaaaiaawIcacaGLPaaaaaa@3ECC@ -1,174 14,111 -27 5,856 -422 5,452 -228 5,974 -1,433 43,923
V ˜ ( F ˜ π ( t ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOvayaaia WaaeWabeaaceWGgbGbaGaadaWgaaWcbaGaeqiWdahabeaakmaabmaa baGaamiDaaGaayjkaiaawMcaaaGaayjkaiaawMcaaaaa@3EE3@ -307 28,421 -460 3,963 -344 3,850 112 4,235 -401 27,987
Table 5.8
Real populations (population size N = 284 ) . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbeqabeWacmGabiqabeqabmqabeabbaGcbaWaaeGaaeaaca WGobGaaGypaiaaikdacaaI4aGaaGinaaGaayzkaaGaaiOlaaaa@3A48@ RBIAS and RRMSE of variance estimators under Poisson sampling with inclusion probabilities proportional to the absolute value of the residuals of the linear regression of the population y i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbeqabeWacmGabiqabeqabmqabeabbaGcbaGaamyEamaaBa aaleaacaWGPbaabeaakiabgkHiTaaa@3807@ values on the population x i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbeqabeWacmGabiqabeqabmqabeabbaGcbaGaamiEamaaBa aaleaacaWGPbaabeaakiabgkHiTaaa@3806@ values. Expected size n * = 30 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbeqabeWacmGabiqabeqabmqabeabbaGcbaGaamOBamaaCa aaleqabaGaaGOkaaaakiaai2dacaaIZaGaaGimaaaa@3914@
Table summary
This table displays the results of Real populations (population size XXXX RBIAS and RRMSE of variance estimators under Poisson sampling with inclusion probabilities proportional to the absolute value of the residuals of the linear regression of the population XXXX values on the population XXXX values. Expected size XXXX XXXX, RBIAS , RRMSE and RRMSE, calculated using MU284 population with XXXX and XXXX and MU284 population with XXXX and XXXX units of measure (appearing as column headers).
  t = F N 1 ( 0.05 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbeqabeWacmGabiqabeqabmqabeabbaGcbaGaamiDaiaai2 dacaWGgbWaa0baaSqaaiaad6eaaeaacqGHsislcaaIXaaaaOWaaeWa aeaacaaIWaGaaGOlaiaaicdacaaI1aaacaGLOaGaayzkaaaaaa@40DB@ t = F N 1 ( 0.25 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbeqabeWacmGabiqabeqabmqabeabbaGcbaGaamiDaiaai2 dacaWGgbWaa0baaSqaaiaad6eaaeaacqGHsislcaaIXaaaaOWaaeWa aeaacaaIWaGaaGOlaiaaikdacaaI1aaacaGLOaGaayzkaaaaaa@40DD@ t = F N 1 ( 0.50 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbeqabeWacmGabiqabeqabmqabeabbaGcbaGaamiDaiaai2 dacaWGgbWaa0baaSqaaiaad6eaaeaacqGHsislcaaIXaaaaOWaaeWa aeaacaaIWaGaaGOlaiaaiwdacaaIWaaacaGLOaGaayzkaaaaaa@40DB@ t = F N 1 ( 0.75 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbeqabeWacmGabiqabeqabmqabeabbaGcbaGaamiDaiaai2 dacaWGgbWaa0baaSqaaiaad6eaaeaacqGHsislcaaIXaaaaOWaaeWa aeaacaaIWaGaaGOlaiaaiEdacaaI1aaacaGLOaGaayzkaaaaaa@40E2@ t = F N 1 ( 0.95 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbeqabeWacmGabiqabeqabmqabeabbaGcbaGaamiDaiaai2 dacaWGgbWaa0baaSqaaiaad6eaaeaacqGHsislcaaIXaaaaOWaaeWa aeaacaaIWaGaaGOlaiaaiMdacaaI1aaacaGLOaGaayzkaaaaaa@40E4@
RBIAS RRMSE RBIAS RRMSE RBIAS RRMSE RBIAS RRMSE RBIAS RRMSE
MU284 population with Y = ln R M T 85 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamywaiaai2 daciGGSbGaaiOBaiaadkfacaWGnbGaamivaiaaiIdacaaI1aaaaa@3EAC@ and X = ln P 85 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiwaiaai2 daciGGSbGaaiOBaiaadcfacaaI4aGaaGynaaaa@3CFE@
V ˜ ( F ˜ s ( t ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOvayaaia WaaeWabeaaceWGgbGbaGaadaWgaaWcbaGaam4CaaqabaGcdaqadaqa aiaadshaaiaawIcacaGLPaaaaiaawIcacaGLPaaaaaa@3E1E@ -3,502 26,342 -1,841 14,037 -2,691 12,087 -3,415 9,674 -5,932 26,823
V ˜ ( F ˜ l ( t ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOvayaaia WaaeWabeaaceWGgbGbaGaadaWgaaWcbaGaamiBaaqabaGcdaqadaqa aiaadshaaiaawIcacaGLPaaaaiaawIcacaGLPaaaaaa@3E17@ -2,159 27,610 -1,782 14,010 -2,840 12,002 -3,186 10,177 -4,455 26,802
V ˜ ( F ˜ s * ( t ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOvayaaia WaaeWabeaaceWGgbGbaGaadaqhaaWcbaGaam4CaaqaaiaaiQcaaaGc daqadaqaaiaadshaaiaawIcacaGLPaaaaiaawIcacaGLPaaaaaa@3ED3@ -434 22,455 515 15,503 -506 31,296 -1,460 23,496 -2,649 78,527
V ˜ ( F ˜ l * ( t ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOvayaaia WaaeWabeaaceWGgbGbaGaadaqhaaWcbaGaamiBaaqaaiaaiQcaaaGc daqadaqaaiaadshaaiaawIcacaGLPaaaaiaawIcacaGLPaaaaaa@3ECC@ -80 22,921 677 15,575 -280 33,294 -1,283 26,612 -1,597 72,166
V ˜ ( F ˜ π ( t ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOvayaaia WaaeWabeaaceWGgbGbaGaadaWgaaWcbaGaeqiWdahabeaakmaabmaa baGaamiDaaGaayjkaiaawMcaaaGaayjkaiaawMcaaaaa@3EE3@ -294 361,991 522 75,891 43 48,764 -241 36,354 90 32,354
  MU284 population with Y = ln R M T 85 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamywaiaai2 daciGGSbGaaiOBaiaadkfacaWGnbGaamivaiaaiIdacaaI1aaaaa@3EAC@ and X = ln R E V 84 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiwaiaai2 daciGGSbGaaiOBaiaadkfacaWGfbGaamOvaiaaiIdacaaI0aaaaa@3EA4@
V ˜ ( F ˜ s ( t ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOvayaaia WaaeWabeaaceWGgbGbaGaadaWgaaWcbaGaam4CaaqabaGcdaqadaqa aiaadshaaiaawIcacaGLPaaaaiaawIcacaGLPaaaaaa@3E1E@ -5,220 18,699 -3,667 8,749 -3,222 7,537 -3,018 9,279 -4,955 44,597
V ˜ ( F ˜ l ( t ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOvayaaia WaaeWabeaaceWGgbGbaGaadaWgaaWcbaGaamiBaaqabaGcdaqadaqa aiaadshaaiaawIcacaGLPaaaaiaawIcacaGLPaaaaaa@3E17@ -4,254 20,765 -3,100 9,180 -3,435 7,231 -3,196 8,540 -3,461 43,206
V ˜ ( F ˜ s * ( t ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOvayaaia WaaeWabeaaceWGgbGbaGaadaqhaaWcbaGaam4CaaqaaiaaiQcaaaGc daqadaqaaiaadshaaiaawIcacaGLPaaaaiaawIcacaGLPaaaaaa@3ED3@ -2,938 18,922 -1,110 11,828 -1,265 8,726 -1,040 10,963 -3,682 89,262
V ˜ ( F ˜ l * ( t ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOvayaaia WaaeWabeaaceWGgbGbaGaadaqhaaWcbaGaamiBaaqaaiaaiQcaaaGc daqadaqaaiaadshaaiaawIcacaGLPaaaaiaawIcacaGLPaaaaaa@3ECC@ -1,938 19,997 -699 12,641 -1,003 9,305 -599 11,545 -1,558 98,798
V ˜ ( F ˜ π ( t ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOvayaaia WaaeWabeaaceWGgbGbaGaadaWgaaWcbaGaeqiWdahabeaakmaabmaa baGaamiDaaGaayjkaiaawMcaaaGaayjkaiaawMcaaaaa@3EE3@ -143 128,401 493 33,934 -255 18,473 -91 17,904 327 16,463

As can be seen from the simulation results, the variance estimators suffer from large variability. This problem is shared by the variance estimator for the Horvitz-Thompson estimator, which occasionally exhibits extremely large RRMSE’s. It is further interesting to note that while the RBIAS of the variance estimators for the generalized difference estimators is almost always negative and at times rather large in absolute value, the RBIAS of the variance estimator for the Horvitz-Thompson estimator is in most of the considered cases positive.

Acknowledgements

This research was partially supported by the FAR 2014-ATE-0200 grant from University of Milano-Bicocca.

Appendix

Let β MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOSdigaaa@36A0@ denote a sequence of real numbers. Throughout this appendix we shall indicate by O i 1 , i 2 , , i k ( β ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4tamaaBa aaleaacaWGPbWaaSbaaWqaaiaaigdaaeqaaSGaaGilaiaadMgadaWg aaadbaGaaGOmaaqabaWccaaISaGaeSOjGSKaaGilaiaadMgadaWgaa adbaGaam4AaaqabaaaleqaaOWaaeWaaeaacqaHYoGyaiaawIcacaGL Paaaaaa@4250@ rest terms that may depend on x i 1 , x i 2 , , x i k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBa aaleaacaWGPbWaaSbaaWqaaiaaigdaaeqaaaWcbeaakiaaiYcacaWG 4bWaaSbaaSqaaiaadMgadaWgaaadbaGaaGOmaaqabaaaleqaaOGaaG ilaiablAciljaaiYcacaWG4bWaaSbaaSqaaiaadMgadaWgaaadbaGa am4Aaaqabaaaleqaaaaa@41AB@ and that are of the same order as the sequence β MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOSdigaaa@36A0@ uniformly for i 1 , i 2 , , i k U . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAamaaBa aaleaacaaIXaaabeaakiaaiYcacaWGPbWaaSbaaSqaaiaaikdaaeqa aOGaaGilaiablAciljaaiYcacaWGPbWaaSbaaSqaaiaadUgaaeqaaO GaeyicI4Saamyvaiaac6caaaa@4126@ Formally, R ( x i 1 , x i 2 , , x i k ) = O i 1 , i 2 , , i k ( β ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOuamaabm aabaGaamiEamaaBaaaleaacaWGPbWaaSbaaWqaaiaaigdaaeqaaaWc beaakiaaiYcacaWG4bWaaSbaaSqaaiaadMgadaWgaaadbaGaaGOmaa qabaaaleqaaOGaaGilaiablAciljaaiYcacaWG4bWaaSbaaSqaaiaa dMgadaWgaaadbaGaam4AaaqabaaaleqaaaGccaGLOaGaayzkaaGaaG ypaiaad+eadaWgaaWcbaGaamyAamaaBaaameaacaaIXaaabeaaliaa iYcacaWGPbWaaSbaaWqaaiaaikdaaeqaaSGaaGilaiablAciljaaiY cacaWGPbWaaSbaaWqaaiaadUgaaeqaaaWcbeaakmaabmaabaGaeqOS digacaGLOaGaayzkaaaaaa@522D@ if

sup i 1 , i 2 , , i k U | R ( x i 1 , x i 2 , , x i k ) | = O ( β ) . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaybuaeqale aacaWGPbWaaSbaaWqaaiaaigdaaeqaaSGaaGilaiaadMgadaWgaaad baGaaGOmaaqabaWccaaISaGaeSOjGSKaaGilaiaadMgadaWgaaadba Gaam4AaaqabaWccqGHiiIZcaaMc8UaamyvaaqabOqaaiGacohacaGG 1bGaaiiCaaaadaabdaqaaiaaykW7caWGsbWaaeWaaeaacaWG4bWaaS baaSqaaiaadMgadaWgaaqaaiaaigdaaeqaaaqabaGccaaISaGaamiE amaaBaaaleaacaWGPbWaaSbaaeaacaaIYaaabeaaaeqaaOGaaGilai ablAciljaaiYcacaWG4bWaaSbaaSqaaiaadMgadaWgaaqaaiaadUga aeqaaaqabaaakiaawIcacaGLPaaacaaMc8oacaGLhWUaayjcSdGaaG ypaiaad+eadaqadaqaaiabek7aIbGaayjkaiaawMcaaiaai6caaaa@5FF4@

Moreover, to simplify the notation, we shall write m i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBamaaBa aaleaacaWGPbaabeaaaaa@370B@ in place of m ( x i ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBamaabm aabaGaamiEamaaBaaaleaacaWGPbaabeaaaOGaayjkaiaawMcaaaaa @399B@ and σ i 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aa0 baaSqaaiaadMgaaeaacaaIYaaaaaaa@3899@ in place of σ 2 ( x i ) . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaW baaSqabeaacaaIYaaaaOWaaeWaaeaacaWG4bWaaSbaaSqaaiaadMga aeqaaaGccaGLOaGaayzkaaGaaiOlaaaa@3C11@

Bias of the model-based Kuo estimator

E ( F ^ ( t ) F N ( t ) ) = E ( 1 N i s j s w i , j [ I ( ε j t m j ) I ( ε i t m i ) ] ) = 1 N i s j s w i , j [ G ( t m j | x j ) G ( t m i | x i ) ] = 1 2 N i s [ G ( 2,0 ) ( t m i | x i ) ( m i ) 2 G ( 1,0 ) ( t m i | x i ) m i ′′ 2 G ( 1,1 ) ( t m i | x i ) m i + G ( 0 , 2 ) ( t m i | x i ) ] j s w i , j ( x j x i ) 2 + o ( λ 2 ) = λ 2 N n N μ 2 2 μ 0 a b [ G ( 2,0 ) ( t m ( x ) | x ) ( m ( x ) ) 2 G ( 1,0 ) ( t m ( x ) | x ) m ′′ ( x ) 2 G ( 1,1 ) ( t m ( x ) | x ) m ( x ) + G ( 0 , 2 ) ( t m ( x ) | x ) ] h s ¯ ( x ) d x + o ( λ 2 ) . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqbaeaabyGaaa aabaGaamyramaabmaabaGabmOrayaajaWaaeWaaeaacaWG0baacaGL OaGaayzkaaGaeyOeI0IaamOramaaBaaaleaacaWGobaabeaakmaabm aabaGaamiDaaGaayjkaiaawMcaaaGaayjkaiaawMcaaaqaaiaai2da caWGfbWaaeWaaeaadaWcaaqaaiaaigdaaeaacaWGobaaamaaqafabe WcbaGaamyAaiabgMGiplaadohaaeqaniabggHiLdGcdaaeqbqaaiaa dEhadaWgaaWcbaGaamyAaiaaiYcacaWGQbaabeaaaeaacaWGQbGaey icI4Saam4Caaqab0GaeyyeIuoakmaadmaabaGaamysamaabmaabaGa eqyTdu2aaSbaaSqaaiaadQgaaeqaaOGaeyizImQaamiDaiabgkHiTi aad2gadaWgaaWcbaGaamOAaaqabaaakiaawIcacaGLPaaacqGHsisl caWGjbWaaeWaaeaacqaH1oqzdaWgaaWcbaGaamyAaaqabaGccqGHKj YOcaWG0bGaeyOeI0IaamyBamaaBaaaleaacaWGPbaabeaaaOGaayjk aiaawMcaaaGaay5waiaaw2faaaGaayjkaiaawMcaaaqaaaqaaiaai2 dadaWcaaqaaiaaigdaaeaacaWGobaaamaaqafabeWcbaGaamyAaiab gMGiplaadohaaeqaniabggHiLdGcdaaeqbqaaiaadEhadaWgaaWcba GaamyAaiaaiYcacaWGQbaabeaaaeaacaWGQbGaeyicI4Saam4Caaqa b0GaeyyeIuoakmaadmaabaGaam4ramaabmaabaWaaqGaaeaacaWG0b GaeyOeI0IaamyBamaaBaaaleaacaWGQbaabeaaaOGaayjcSdGaamiE amaaBaaaleaacaWGQbaabeaaaOGaayjkaiaawMcaaiabgkHiTiaadE eadaqadaqaamaaeiaabaGaamiDaiabgkHiTiaad2gadaWgaaWcbaGa amyAaaqabaaakiaawIa7aiaadIhadaWgaaWcbaGaamyAaaqabaaaki aawIcacaGLPaaaaiaawUfacaGLDbaaaeaaaeaacaaI9aWaaSaaaeaa caaIXaaabaGaaGOmaiaad6eaaaWaaabuaeqaleaacaWGPbGaeyycI8 Saam4Caaqab0GaeyyeIuoakmaadeaabaGaam4ramaaCaaaleqabaWa aeWaaeaacaaIYaGaaGilaiaaicdaaiaawIcacaGLPaaaaaGcdaqada qaamaaeiaabaGaamiDaiabgkHiTiaad2gadaWgaaWcbaGaamyAaaqa baaakiaawIa7aiaadIhadaWgaaWcbaGaamyAaaqabaaakiaawIcaca GLPaaadaqadaqaaiqad2gagaqbamaaBaaaleaacaWGPbaabeaaaOGa ayjkaiaawMcaamaaCaaaleqabaGaaGOmaaaakiabgkHiTiaadEeada ahaaWcbeqaamaabmaabaGaaGymaiaaiYcacaaIWaaacaGLOaGaayzk aaaaaOWaaeWaaeaadaabcaqaaiaadshacqGHsislcaWGTbWaaSbaaS qaaiaadMgaaeqaaaGccaGLiWoacaWG4bWaaSbaaSqaaiaadMgaaeqa aaGccaGLOaGaayzkaaGabmyBayaagaWaaSbaaSqaaiaadMgaaeqaaa GccaGLBbaaaeaaaeaacaaMe8UaaGjbVpaadiaabaGaeyOeI0IaaGOm aiaadEeadaahaaWcbeqaamaabmaabaGaaGymaiaaiYcacaaIXaaaca GLOaGaayzkaaaaaOWaaeWaaeaadaabcaqaaiaadshacqGHsislcaWG TbWaaSbaaSqaaiaadMgaaeqaaaGccaGLiWoacaWG4bWaaSbaaSqaai aadMgaaeqaaaGccaGLOaGaayzkaaGabmyBayaafaWaaSbaaSqaaiaa dMgaaeqaaOGaey4kaSIaam4ramaaCaaaleqabaWaaeWaaeaacaaIWa GaaiilaiaaikdaaiaawIcacaGLPaaaaaGcdaqadaqaamaaeiaabaGa amiDaiabgkHiTiaad2gadaWgaaWcbaGaamyAaaqabaaakiaawIa7ai aadIhadaWgaaWcbaGaamyAaaqabaaakiaawIcacaGLPaaaaiaaw2fa amaaqafabaGaam4DamaaBaaaleaacaWGPbGaaGilaiaadQgaaeqaaO WaaeWaaeaacaWG4bWaaSbaaSqaaiaadQgaaeqaaOGaeyOeI0IaamiE amaaBaaaleaacaWGPbaabeaaaOGaayjkaiaawMcaamaaCaaaleqaba GaaGOmaaaaaeaacaWGQbGaeyicI4Saam4Caaqab0GaeyyeIuoakiab gUcaRiaad+gadaqadaqaaiabeU7aSnaaCaaaleqabaGaaGOmaaaaaO GaayjkaiaawMcaaaqaaaqaaiaai2dacqaH7oaBdaahaaWcbeqaaiaa ikdaaaGcdaWcaaqaaiaad6eacqGHsislcaWGUbaabaGaamOtaaaada WcaaqaaiabeY7aTnaaBaaaleaacaaIYaaabeaaaOqaaiaaikdacqaH 8oqBdaWgaaWcbaGaaGimaaqabaaaaOWaa8qmaeqaleaacaWGHbaaba GaamOyaaqdcqGHRiI8aOWaamqaaeaacaWGhbWaaWbaaSqabeaadaqa daqaaiaaikdacaaISaGaaGimaaGaayjkaiaawMcaaaaakmaabmaaba WaaqGaaeaacaWG0bGaeyOeI0IaamyBamaabmaabaGaamiEaaGaayjk aiaawMcaaiaaykW7aiaawIa7aiaadIhaaiaawIcacaGLPaaadaqada qaaiqad2gagaqbamaabmaabaGaamiEaaGaayjkaiaawMcaaaGaayjk aiaawMcaamaaCaaaleqabaGaaGOmaaaakiabgkHiTiaadEeadaahaa WcbeqaamaabmaabaGaaGymaiaaiYcacaaIWaaacaGLOaGaayzkaaaa aOWaaeWaaeaadaabcaqaaiaadshacqGHsislcaWGTbWaaeWaaeaaca WG4baacaGLOaGaayzkaaGaaGPaVdGaayjcSdGaamiEaaGaayjkaiaa wMcaaiqad2gagaqbgaqbamaabmaabaGaamiEaaGaayjkaiaawMcaaa Gaay5waaaabaaabaGaaGjbVlaaysW7daWacaqaaiabgkHiTiaaikda caWGhbWaaWbaaSqabeaadaqadaqaaiaaigdacaaISaGaaGymaaGaay jkaiaawMcaaaaakmaabmaabaWaaqGaaeaacaWG0bGaeyOeI0IaamyB amaabmaabaGaamiEaaGaayjkaiaawMcaaiaaykW7aiaawIa7aiaadI haaiaawIcacaGLPaaaceWGTbGbauaadaqadaqaaiaadIhaaiaawIca caGLPaaacqGHRaWkcaWGhbWaaWbaaSqabeaadaqadaqaaiaaicdaca GGSaGaaGOmaaGaayjkaiaawMcaaaaakmaabmaabaWaaqGaaeaacaWG 0bGaeyOeI0IaamyBamaabmaabaGaamiEaaGaayjkaiaawMcaaiaayk W7aiaawIa7aiaadIhaaiaawIcacaGLPaaaaiaaw2faaiaadIgadaWg aaWcbaGaaGPaVlqadohagaqeaaqabaGcdaqadaqaaiaadIhaaiaawI cacaGLPaaacaWGKbGaamiEaiabgUcaRiaad+gadaqadaqaaiabeU7a SnaaCaaaleqabaGaaGOmaaaaaOGaayjkaiaawMcaaiaai6caaaaaaa@71E0@

Bias of the generalized difference Kuo estimator

Write

F ˜ ( t ) F N ( t ) = 1 N { i s j s w ˜ i , j [ I ( ε j t m j ) I ( ε i t m i ) ] + i s ( 1 1 π i ) j s w ˜ i , j [ I ( ε j t m j ) I ( ε i t m i ) ] } . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrVeFfea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peee0hXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaafaqaaeGaca aabaGabmOrayaaiaWaaeWaaeaacaWG0baacaGLOaGaayzkaaGaeyOe I0IaamOramaaBaaaleaacaWGobaabeaakmaabmaabaGaamiDaaGaay jkaiaawMcaaaqaaiaai2dadaWcaaqaaiaaigdaaeaacaWGobaaamaa ceaabaWaaabuaeqaleaacaWGPbGaeyycI8Saam4Caaqab0GaeyyeIu oakmaaqafabaGabm4DayaaiaWaaSbaaSqaaiaadMgacaaISaGaamOA aaqabaaabaGaamOAaiabgIGiolaadohaaeqaniabggHiLdGcdaWada qaaiaadMeadaqadaqaaiabew7aLnaaBaaaleaacaWGQbaabeaakiab gsMiJkaadshacqGHsislcaWGTbWaaSbaaSqaaiaadQgaaeqaaaGcca GLOaGaayzkaaGaeyOeI0IaamysamaabmaabaGaeqyTdu2aaSbaaSqa aiaadMgaaeqaaOGaeyizImQaamiDaiabgkHiTiaad2gadaWgaaWcba GaamyAaaqabaaakiaawIcacaGLPaaaaiaawUfacaGLDbaaaiaawUha aaqaaaqaaiaaysW7caaMe8+aaiGaaeaacaaMe8UaaGjbVlaaysW7ca aMe8UaaGjbVlaaysW7cqGHRaWkdaaeqbqabSqaaiaadMgacqGHiiIZ caWGZbaabeqdcqGHris5aOWaaeWaaeaacaaIXaGaeyOeI0YaaSaaae aacaaIXaaabaGaeqiWda3aaSbaaSqaaiaadMgaaeqaaaaaaOGaayjk aiaawMcaaiaaysW7daaeqbqaaiqadEhagaacamaaBaaaleaacaWGPb GaaGilaiaadQgaaeqaaaqaaiaadQgacqGHiiIZcaWGZbaabeqdcqGH ris5aOWaamWaaeaacaWGjbWaaeWaaeaacqaH1oqzdaWgaaWcbaGaam OAaaqabaGccqGHKjYOcaWG0bGaeyOeI0IaamyBamaaBaaaleaacaWG QbaabeaaaOGaayjkaiaawMcaaiabgkHiTiaadMeadaqadaqaaiabew 7aLnaaBaaaleaacaWGPbaabeaakiabgsMiJkaadshacqGHsislcaWG TbWaaSbaaSqaaiaadMgaaeqaaaGccaGLOaGaayzkaaaacaGLBbGaay zxaaaacaGL9baacaaIUaaaaaaa@AADE@

Similar steps as those seen for F ^ ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaja WaaeWaaeaacaWG0baacaGLOaGaayzkaaaaaa@385C@ show that

E ( F ˜ ( t ) F N ( t ) ) = λ 2 N n N μ 2 2 μ 0 a b [ G ( 2,0 ) ( t m ( x ) | x ) ( m ( x ) ) 2 G ( 1,0 ) ( t m ( x ) | x ) m ′′ ( x ) 2 G ( 1,1 ) ( t m ( x ) | x ) m ( x ) + G ( 0 , 2 ) ( t m ( x ) | x ) ] h ( x ) d x + o ( λ 2 ) , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peee0hXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaafaqaaeGaca aabaGaamyramaabmaabaGabmOrayaaiaWaaeWaaeaacaWG0baacaGL OaGaayzkaaGaeyOeI0IaamOramaaBaaaleaacaWGobaabeaakmaabm aabaGaamiDaaGaayjkaiaawMcaaaGaayjkaiaawMcaaaqaaiaai2da cqaH7oaBdaahaaWcbeqaaiaaikdaaaGcdaWcaaqaaiaad6eacqGHsi slcaWGUbaabaGaamOtaaaadaWcaaqaaiabeY7aTnaaBaaaleaacaaI YaaabeaaaOqaaiaaikdacqaH8oqBdaWgaaWcbaGaaGimaaqabaaaaO Waa8qmaeaadaWabaqaaiaadEeadaahaaWcbeqaamaabmaabaGaaGOm aiaaiYcacaaIWaaacaGLOaGaayzkaaaaaOWaaeWaaeaadaabcaqaai aadshacqGHsislcaWGTbWaaeWaaeaacaWG4baacaGLOaGaayzkaaGa aGPaVdGaayjcSdGaamiEaaGaayjkaiaawMcaamaabmaabaGabmyBay aafaWaaeWaaeaacaWG4baacaGLOaGaayzkaaaacaGLOaGaayzkaaWa aWbaaSqabeaacaaIYaaaaOGaeyOeI0Iaam4ramaaCaaaleqabaWaae WaaeaacaaIXaGaaGilaiaaicdaaiaawIcacaGLPaaaaaGcdaqadaqa amaaeiaabaGaamiDaiabgkHiTiaad2gadaqadaqaaiaadIhaaiaawI cacaGLPaaacaaMc8oacaGLiWoacaWG4baacaGLOaGaayzkaaGabmyB ayaafyaafaWaaeWaaeaacaWG4baacaGLOaGaayzkaaaacaGLBbaaaS qaaiaadggaaeaacaWGIbaaniabgUIiYdaakeaaaeaadaWacaqaaiaa ysW7caaMe8UaaGjbVlaaysW7caaMe8UaaGjbVlaaysW7caaMe8UaaG jbVlaaysW7caaMe8UaaGjbVlaaysW7caaMe8UaaGjbVlaaysW7caaM e8UaaGjbVlaaysW7caaMe8UaeyOeI0IaaGOmaiaadEeadaahaaWcbe qaamaabmaabaGaaGymaiaaiYcacaaIXaaacaGLOaGaayzkaaaaaOWa aeWaaeaadaabcaqaaiaadshacqGHsislcaWGTbWaaeWaaeaacaWG4b aacaGLOaGaayzkaaGaaGPaVdGaayjcSdGaamiEaaGaayjkaiaawMca aiqad2gagaqbamaabmaabaGaamiEaaGaayjkaiaawMcaaiabgUcaRi aadEeadaahaaWcbeqaamaabmaabaGaaGimaiaacYcacaaIYaaacaGL OaGaayzkaaaaaOWaaeWaaeaadaabcaqaaiaadshacqGHsislcaWGTb WaaeWaaeaacaWG4baacaGLOaGaayzkaaGaaGPaVdGaayjcSdGaamiE aaGaayjkaiaawMcaaaGaayzxaaGaamiAamaabmaabaGaamiEaaGaay jkaiaawMcaaiaadsgacaWG4bGaey4kaSIaam4BamaabmaabaGaeq4U dW2aaWbaaSqabeaacaaIYaaaaaGccaGLOaGaayzkaaGaaiilaaaaaa a@D090@

where

h ( x ) := h s ¯ ( x ) + ( 1 π 1 ( x ) ) h s ( x ) . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiAamaabm aabaGaamiEaaGaayjkaiaawMcaaiaaiQdacaaI9aGaamiAamaaBaaa leaacaaMc8Uabm4CayaaraaabeaakmaabmaabaGaamiEaaGaayjkai aawMcaaiabgUcaRmaabmaabaGaaGymaiabgkHiTiabec8aWnaaCaaa leqabaGaeyOeI0IaaGymaaaakmaabmaabaGaamiEaaGaayjkaiaawM caaaGaayjkaiaawMcaaiaadIgadaWgaaWcbaGaam4CaaqabaGcdaqa daqaaiaadIhaaiaawIcacaGLPaaacaaIUaaaaa@4FCE@

Variance of the model-based Kuo estimator

var ( F ^ ( t ) F N ( t ) ) = var ( 1 N i s j s w i , j I ( ε j t m j ) 1 N i s I ( y i t ) ) = 1 N 2 i 1 s i 2 s j s w i 1 , j w i 2 , j [ G ( t m j | x j ) G 2 ( t m j | x j ) ] + 1 N 2 i s [ G ( t m i | x i ) G 2 ( t m i | x i ) ] = A 1 + A 2 , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrViFD0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqbaeaabqGaaa aabaGaaeODaiaabggacaqGYbWaaeWaaeaaceWGgbGbaKaadaqadaqa aiaadshaaiaawIcacaGLPaaacqGHsislcaWGgbWaaSbaaSqaaiaad6 eaaeqaaOWaaeWaaeaacaWG0baacaGLOaGaayzkaaaacaGLOaGaayzk aaaabaGaaGypaiaabAhacaqGHbGaaeOCamaabmaabaWaaSaaaeaaca aIXaaabaGaamOtaaaadaaeqbqabSqaaiaadMgacqGHjiYZcaWGZbaa beqdcqGHris5aOWaaabuaeaacaWG3bWaaSbaaSqaaiaadMgacaaISa GaamOAaaqabaGccaWGjbaaleaacaWGQbGaeyicI4Saam4Caaqab0Ga eyyeIuoakmaabmaabaGaeqyTdu2aaSbaaSqaaiaadQgaaeqaaOGaey izImQaamiDaiabgkHiTiaad2gadaWgaaWcbaGaamOAaaqabaaakiaa wIcacaGLPaaacqGHsisldaWcaaqaaiaaigdaaeaacaWGobaaamaaqa fabaGaamysamaabmaabaGaamyEamaaBaaaleaacaWGPbaabeaakiab gsMiJkaadshaaiaawIcacaGLPaaaaSqaaiaadMgacqGHjiYZcaWGZb aabeqdcqGHris5aaGccaGLOaGaayzkaaaabaaabaGaaGypamaalaaa baGaaGymaaqaaiaad6eadaahaaWcbeqaaiaaikdaaaaaaOWaaabuae aadaaeqbqaamaaqafabaGaam4DamaaBaaaleaacaWGPbWaaSbaaWqa aiaaigdaaeqaaSGaaGilaiaadQgaaeqaaOGaam4DamaaBaaaleaaca WGPbWaaSbaaWqaaiaaikdaaeqaaSGaaGilaiaadQgaaeqaaaqaaiaa dQgacqGHiiIZcaWGZbaabeqdcqGHris5aaWcbaGaamyAamaaBaaame aacaaIYaaabeaaliabgMGiplaadohaaeqaniabggHiLdaaleaacaWG PbWaaSbaaWqaaiaaigdaaeqaaSGaeyycI8Saam4Caaqab0GaeyyeIu oakmaadmaabaGaam4ramaabmaabaWaaqGaaeaacaWG0bGaeyOeI0Ia amyBamaaBaaaleaacaWGQbaabeaaaOGaayjcSdGaaGPaVlaadIhada WgaaWcbaGaamOAaaqabaaakiaawIcacaGLPaaacqGHsislcaWGhbWa aWbaaSqabeaacaaIYaaaaOWaaeWaaeaadaabcaqaaiaadshacqGHsi slcaWGTbWaaSbaaSqaaiaadQgaaeqaaaGccaGLiWoacaaMc8UaamiE amaaBaaaleaacaWGQbaabeaaaOGaayjkaiaawMcaaaGaay5waiaaw2 faaaqaaaqaaiaaysW7caaMe8Uaey4kaSYaaSaaaeaacaaIXaaabaGa amOtamaaCaaaleqabaGaaGOmaaaaaaGcdaaeqbqabSqaaiaadMgacq GHjiYZcaWGZbaabeqdcqGHris5aOWaamWaaeaacaWGhbWaaeWaaeaa daabcaqaaiaadshacqGHsislcaWGTbWaaSbaaSqaaiaadMgaaeqaaa GccaGLiWoacaaMc8UaamiEamaaBaaaleaacaWGPbaabeaaaOGaayjk aiaawMcaaiabgkHiTiaadEeadaahaaWcbeqaaiaaikdaaaGcdaqada qaamaaeiaabaGaamiDaiabgkHiTiaad2gadaWgaaWcbaGaamyAaaqa baaakiaawIa7aiaaykW7caWG4bWaaSbaaSqaaiaadMgaaeqaaaGcca GLOaGaayzkaaaacaGLBbGaayzxaaaabaaabaGaaGypaiaadgeadaWg aaWcbaGaaGymaaqabaGccqGHRaWkcaWGbbWaaSbaaSqaaiaaikdaae qaaOGaaGilaaaaaaa@D979@

where

A 1 := 1 N 2 i 1 s i 2 s j s w i 1 , j w i 2 , j [ G ( t m j | x j ) G 2 ( t m j | x j ) ] = 1 N 2 j s [ G ( t m j | x j ) G 2 ( t m j | x j ) ] ( i s w i , j ) 2 = 1 n ( N n N ) 2 a b [ G ( t m ( x ) | x ) G 2 ( t m ( x ) | x ) ] [ h s ¯ ( x ) / h s ( x ) ] h s ¯ ( x ) d x + O ( ( n λ ) 1 α ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrViFD0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqbaeaabqGaaa aabaGaamyqamaaBaaaleaacaaIXaaabeaaaOqaaiaaiQdacaaI9aWa aSaaaeaacaaIXaaabaGaamOtamaaCaaaleqabaGaaGOmaaaaaaGcda aeqbqaamaaqafabaWaaabuaeaacaWG3bWaaSbaaSqaaiaadMgadaWg aaadbaGaaGymaaqabaWccaaISaGaamOAaaqabaGccaWG3bWaaSbaaS qaaiaadMgadaWgaaadbaGaaGOmaaqabaWccaaISaGaamOAaaqabaaa baGaamOAaiabgIGiolaadohaaeqaniabggHiLdaaleaacaWGPbWaaS baaWqaaiaaikdaaeqaaSGaeyycI8Saam4Caaqab0GaeyyeIuoaaSqa aiaadMgadaWgaaadbaGaaGymaaqabaWccqGHjiYZcaWGZbaabeqdcq GHris5aOWaamWaaeaacaWGhbWaaeWaaeaadaabcaqaaiaadshacqGH sislcaWGTbWaaSbaaSqaaiaadQgaaeqaaaGccaGLiWoacaaMc8Uaam iEamaaBaaaleaacaWGQbaabeaaaOGaayjkaiaawMcaaiabgkHiTiaa dEeadaahaaWcbeqaaiaaikdaaaGcdaqadaqaamaaeiaabaGaamiDai abgkHiTiaad2gadaWgaaWcbaGaamOAaaqabaaakiaawIa7aiaaykW7 caWG4bWaaSbaaSqaaiaadQgaaeqaaaGccaGLOaGaayzkaaaacaGLBb GaayzxaaaabaaabaGaaGypamaalaaabaGaaGymaaqaaiaad6eadaah aaWcbeqaaiaaikdaaaaaaOWaaabuaeqaleaacaWGQbGaeyicI4Saam 4Caaqab0GaeyyeIuoakmaadmaabaGaam4ramaabmaabaWaaqGaaeaa caWG0bGaeyOeI0IaamyBamaaBaaaleaacaWGQbaabeaaaOGaayjcSd GaaGPaVlaadIhadaWgaaWcbaGaamOAaaqabaaakiaawIcacaGLPaaa cqGHsislcaWGhbWaaWbaaSqabeaacaaIYaaaaOWaaeWaaeaadaabca qaaiaadshacqGHsislcaWGTbWaaSbaaSqaaiaadQgaaeqaaaGccaGL iWoacaaMc8UaamiEamaaBaaaleaacaWGQbaabeaaaOGaayjkaiaawM caaaGaay5waiaaw2faamaabmaabaWaaabuaeqaleaacaWGPbGaeyyc I8Saam4Caaqab0GaeyyeIuoakiaadEhadaWgaaWcbaGaamyAaiaaiY cacaWGQbaabeaaaOGaayjkaiaawMcaamaaCaaaleqabaGaaGOmaaaa aOqaaaqaaiaai2dadaWcaaqaaiaaigdaaeaacaWGUbaaamaabmaaba WaaSaaaeaacaWGobGaeyOeI0IaamOBaaqaaiaad6eaaaaacaGLOaGa ayzkaaWaaWbaaSqabeaacaaIYaaaaOWaa8qmaeqaleaacaWGHbaaba GaamOyaaqdcqGHRiI8aOWaamWaaeaacaWGhbWaaeWaaeaadaabcaqa aiaadshacqGHsislcaWGTbWaaeWaaeaacaWG4baacaGLOaGaayzkaa GaaGPaVdGaayjcSdGaaGPaVlaadIhaaiaawIcacaGLPaaacqGHsisl caWGhbWaaWbaaSqabeaacaaIYaaaaOWaaeWaaeaadaabcaqaaiaads hacqGHsislcaWGTbWaaeWaaeaacaWG4baacaGLOaGaayzkaaGaaGPa VdGaayjcSdGaaGPaVlaadIhaaiaawIcacaGLPaaaaiaawUfacaGLDb aadaWadaqaamaalyaabaGaamiAamaaBaaaleaacaaMc8Uabm4Cayaa raaabeaakmaabmaabaGaamiEaaGaayjkaiaawMcaaaqaaiaadIgada WgaaWcbaGaam4CaaqabaGcdaqadaqaaiaadIhaaiaawIcacaGLPaaa aaaacaGLBbGaayzxaaGaamiAamaaBaaaleaacaaMc8Uabm4Cayaara aabeaakmaabmaabaGaamiEaaGaayjkaiaawMcaaiaadsgacaWG4baa baaabaGaaGjbVlaaysW7cqGHRaWkcaWGpbWaaeWaaeaadaqadaqaai aad6gacqaH7oaBaiaawIcacaGLPaaadaahaaWcbeqaaiabgkHiTiaa igdaaaGccqaHXoqyaiaawIcacaGLPaaaaaaaaa@F03D@

and

A 2 := 1 N 2 i s [ G ( t m i | x i ) G 2 ( t m i | x i ) ] = 1 N n ( N n N ) 2 a b [ G ( t m ( x ) | x ) G 2 ( t m ( x ) | x ) ] h s ¯ ( x ) d x + O ( n 1 α ) . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrViFD0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqbaeaabiGaaa qaaiaadgeadaWgaaWcbaGaaGOmaaqabaaakeaacaaI6aGaaGypamaa laaabaGaaGymaaqaaiaad6eadaahaaWcbeqaaiaaikdaaaaaaOWaaa buaeqaleaacaWGPbGaeyycI8Saam4Caaqab0GaeyyeIuoakmaadmaa baGaam4ramaabmaabaWaaqGaaeaacaWG0bGaeyOeI0IaamyBamaaBa aaleaacaWGPbaabeaaaOGaayjcSdGaaGPaVlaadIhadaWgaaWcbaGa amyAaaqabaaakiaawIcacaGLPaaacqGHsislcaWGhbWaaWbaaSqabe aacaaIYaaaaOWaaeWaaeaadaabcaqaaiaadshacqGHsislcaWGTbWa aSbaaSqaaiaadMgaaeqaaaGccaGLiWoacaaMc8UaamiEamaaBaaale aacaWGPbaabeaaaOGaayjkaiaawMcaaaGaay5waiaaw2faaaqaaaqa aiaai2dadaWcaaqaaiaaigdaaeaacaWGobGaeyOeI0IaamOBaaaada qadaqaamaalaaabaGaamOtaiabgkHiTiaad6gaaeaacaWGobaaaaGa ayjkaiaawMcaamaaCaaaleqabaGaaGOmaaaakmaapedabeWcbaGaam yyaaqaaiaadkgaa0Gaey4kIipakmaadmaabaGaam4ramaabmaabaWa aqGaaeaacaWG0bGaeyOeI0IaamyBamaabmaabaGaamiEaaGaayjkai aawMcaaiaaykW7aiaawIa7aiaaykW7caWG4baacaGLOaGaayzkaaGa eyOeI0Iaam4ramaaCaaaleqabaGaaGOmaaaakmaabmaabaWaaqGaae aacaWG0bGaeyOeI0IaamyBamaabmaabaGaamiEaaGaayjkaiaawMca aiaaykW7aiaawIa7aiaaykW7caWG4baacaGLOaGaayzkaaaacaGLBb GaayzxaaGaamiAamaaBaaaleaacaaMc8Uabm4Cayaaraaabeaakmaa bmaabaGaamiEaaGaayjkaiaawMcaaiaadsgacaWG4bGaey4kaSIaam 4tamaabmaabaGaamOBamaaCaaaleqabaGaeyOeI0IaaGymaaaakiab eg7aHbGaayjkaiaawMcaaiaai6caaaaaaa@9941@

Thus,

var ( F ^ ( t ) F N ( t ) ) = 1 n ( N n N ) 2 a b [ G ( t m ( x ) | x ) G 2 ( t m ( x ) | x ) ] [ h s ¯ ( x ) / h s ( x ) ] h s ¯ ( x ) d x + 1 N n ( N n N ) 2 a b [ G ( t m ( x ) | x ) G 2 ( t m ( x ) | x ) ] h s ¯ ( x ) d x + O ( ( n λ ) 1 α ) . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqbaeaabiGaaa qaaiaabAhacaqGHbGaaeOCamaabmaabaGabmOrayaajaWaaeWaaeaa caWG0baacaGLOaGaayzkaaGaeyOeI0IaamOramaaBaaaleaacaWGob aabeaakmaabmaabaGaamiDaaGaayjkaiaawMcaaaGaayjkaiaawMca aaqaaiabg2da9maalaaabaGaaGymaaqaaiaad6gaaaWaaeWaaeaada Wcaaqaaiaad6eacqGHsislcaWGUbaabaGaamOtaaaaaiaawIcacaGL PaaadaahaaWcbeqaaiaaikdaaaGcdaWdXaqabSqaaiaadggaaeaaca WGIbaaniabgUIiYdGcdaWadaqaaiaadEeadaqadaqaamaaeiaabaGa amiDaiabgkHiTiaad2gadaqadaqaaiaadIhaaiaawIcacaGLPaaaca aMc8oacaGLiWoacaaMe8UaamiEaaGaayjkaiaawMcaaiabgkHiTiaa dEeadaahaaWcbeqaaiaaikdaaaGcdaqadaqaamaaeiaabaGaamiDai abgkHiTiaad2gadaqadaqaaiaadIhaaiaawIcacaGLPaaacaaMc8oa caGLiWoacaaMe8UaamiEaaGaayjkaiaawMcaaaGaay5waiaaw2faam aadmaabaWaaSGbaeaacaWGObWaaSbaaSqaaiaaykW7ceWGZbGbaeba aeqaaOWaaeWaaeaacaWG4baacaGLOaGaayzkaaaabaGaamiAamaaBa aaleaacaWGZbaabeaakmaabmaabaGaamiEaaGaayjkaiaawMcaaaaa aiaawUfacaGLDbaacaWGObWaaSbaaSqaaiaaykW7ceWGZbGbaebaae qaaOWaaeWaaeaacaWG4baacaGLOaGaayzkaaGaamizaiaadIhaaeaa aeaacqGHRaWkdaWcaaqaaiaaigdaaeaacaWGobGaeyOeI0IaamOBaa aadaqadaqaamaalaaabaGaamOtaiabgkHiTiaad6gaaeaacaWGobaa aaGaayjkaiaawMcaamaaCaaaleqabaGaaGOmaaaakmaapedabeWcba Gaamyyaaqaaiaadkgaa0Gaey4kIipakmaadmaabaGaam4ramaabmaa baWaaqGaaeaacaWG0bGaeyOeI0IaamyBamaabmaabaGaamiEaaGaay jkaiaawMcaaiaaykW7aiaawIa7aiaaysW7caWG4baacaGLOaGaayzk aaGaeyOeI0Iaam4ramaaCaaaleqabaGaaGOmaaaakmaabmaabaWaaq GaaeaacaWG0bGaeyOeI0IaamyBamaabmaabaGaamiEaaGaayjkaiaa wMcaaiaaykW7aiaawIa7aiaaysW7caWG4baacaGLOaGaayzkaaaaca GLBbGaayzxaaGaamiAamaaBaaaleaacaaMc8Uabm4Cayaaraaabeaa kmaabmaabaGaamiEaaGaayjkaiaawMcaaiaadsgacaWG4bGaey4kaS Iaam4tamaabmaabaWaaeWaaeaacaWGUbGaeq4UdWgacaGLOaGaayzk aaWaaWbaaSqabeaacqGHsislcaaIXaaaaOGaeqySdegacaGLOaGaay zkaaGaaGOlaaaaaaa@C354@

Variance of the generalized difference Kuo estimator

Note that

F ˜ ( t ) F N ( t ) = 1 N { j s I ( y j t ) [ i s w ˜ i , j i s w ˜ i , j ( π i 1 1 ) + ( π j 1 1 ) ] i s I ( y i t ) } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaia WaaeWaaeaacaWG0baacaGLOaGaayzkaaGaeyOeI0IaamOramaaBaaa leaacaWGobaabeaakmaabmaabaGaamiDaaGaayjkaiaawMcaaiaai2 dadaWcaaqaaiaaigdaaeaacaWGobaaamaacmaabaWaaabuaeaacaWG jbWaaeWaaeaacaWG5bWaaSbaaSqaaiaadQgaaeqaaOGaeyizImQaam iDaaGaayjkaiaawMcaaaWcbaGaamOAaiabgIGiolaadohaaeqaniab ggHiLdGcdaWadaqaamaaqafabaGabm4DayaaiaWaaSbaaSqaaiaadM gacaaISaGaamOAaaqabaaabaGaamyAaiabgMGiplaadohaaeqaniab ggHiLdGccqGHsisldaaeqbqaaiqadEhagaacamaaBaaaleaacaWGPb GaaGilaiaadQgaaeqaaaqaaiaadMgacqGHiiIZcaWGZbaabeqdcqGH ris5aOWaaeWaaeaacqaHapaCdaqhaaWcbaGaamyAaaqaaiabgkHiTi aaigdaaaGccqGHsislcaaIXaaacaGLOaGaayzkaaGaey4kaSYaaeWa aeaacqaHapaCdaqhaaWcbaGaamOAaaqaaiabgkHiTiaaigdaaaGccq GHsislcaaIXaaacaGLOaGaayzkaaaacaGLBbGaayzxaaGaeyOeI0Ya aabuaeaacaWGjbWaaeWaaeaacaWG5bWaaSbaaSqaaiaadMgaaeqaaO GaeyizImQaamiDaaGaayjkaiaawMcaaaWcbaGaamyAaiabgMGiplaa dohaaeqaniabggHiLdaakiaawUhacaGL9baaaaa@8286@

so that

var ( F ˜ ( t ) F N ( t ) ) = var ( 1 N j s I ( y j t ) [ i s w ˜ i , j + ( π j 1 1 ) i s w ˜ i , j ( π i 1 1 ) ] ) + var ( 1 N i s I ( y i t ) ) = B 1 + A 2 , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqbaeaabmGaaa qaaiaabAhacaqGHbGaaeOCamaabmaabaGabmOrayaaiaWaaeWaaeaa caWG0baacaGLOaGaayzkaaGaeyOeI0IaamOramaaBaaaleaacaWGob aabeaakmaabmaabaGaamiDaaGaayjkaiaawMcaaaGaayjkaiaawMca aaqaaiaai2dacaqG2bGaaeyyaiaabkhadaqadaqaamaalaaabaGaaG ymaaqaaiaad6eaaaWaaabuaeaacaWGjbWaaeWaaeaacaWG5bWaaSba aSqaaiaadQgaaeqaaOGaeyizImQaamiDaaGaayjkaiaawMcaaaWcba GaamOAaiabgIGiolaadohaaeqaniabggHiLdGcdaWadaqaamaaqafa baGabm4DayaaiaWaaSbaaSqaaiaadMgacaaISaGaamOAaaqabaaaba GaamyAaiabgMGiplaadohaaeqaniabggHiLdGccqGHRaWkdaqadaqa aiabec8aWnaaDaaaleaacaWGQbaabaGaeyOeI0IaaGymaaaakiabgk HiTiaaigdaaiaawIcacaGLPaaacqGHsisldaaeqbqaaiqadEhagaac amaaBaaaleaacaWGPbGaaGilaiaadQgaaeqaaaqaaiaadMgacqGHii IZcaWGZbaabeqdcqGHris5aOWaaeWaaeaacqaHapaCdaqhaaWcbaGa amyAaaqaaiabgkHiTiaaigdaaaGccqGHsislcaaIXaaacaGLOaGaay zkaaaacaGLBbGaayzxaaaacaGLOaGaayzkaaaabaaabaGaaGjbVlaa ysW7cqGHRaWkcaqG2bGaaeyyaiaabkhadaqadaqaamaalaaabaGaaG ymaaqaaiaad6eaaaWaaabuaeaacaWGjbWaaeWaaeaacaWG5bWaaSba aSqaaiaadMgaaeqaaOGaeyizImQaamiDaaGaayjkaiaawMcaaaWcba GaamyAaiabgMGiplaadohaaeqaniabggHiLdaakiaawIcacaGLPaaa aeaaaeaacaaI9aGaamOqamaaBaaaleaacaaIXaaabeaakiabgUcaRi aadgeadaWgaaWcbaGaaGOmaaqabaGccaaISaaaaaaa@97F4@

where A 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqamaaBa aaleaacaaIYaaabeaaaaa@36AD@ is the same as in the variance of F ^ ( t ) , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaja WaaeWaaeaacaWG0baacaGLOaGaayzkaaGaaiilaaaa@390C@ and where

B 1 := var ( 1 N j s I ( y j t ) [ i s w ˜ i , j + ( π j 1 1 ) i s w ˜ i , j ( π i 1 1 ) ] ) = 1 N 2 j s [ G ( t m j | x j ) G 2 ( t m j | x j ) ] [ i s w ˜ i , j + ( π j 1 1 ) i s w ˜ i , j ( π i 1 1 ) ] 2 = 1 N 2 j s [ G ( t m j | x j ) G 2 ( t m j | x j ) ] [ i s w ˜ i , j + ( π j 1 1 ) ( 1 i s w ˜ i , j ) ] 2 + O ( λ n 1 ) = 1 n ( N n N ) 2 a b [ G ( t m ( x ) | x ) G 2 ( t m ( x ) | x ) ] [ h s ¯ ( x ) / h s ( x ) ] h s ¯ ( x ) d x + O ( ( n λ ) 1 α + λ n 1 ) = A 1 + O ( ( n λ ) 1 α + λ n 1 ) . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqbaeaabyGaaa aabaGaamOqamaaBaaaleaacaaIXaaabeaaaOqaaiaaiQdacaaI9aGa aeODaiaabggacaqGYbWaaeWaaeaadaWcaaqaaiaaigdaaeaacaWGob aaamaaqafabaGaamysamaabmaabaGaamyEamaaBaaaleaacaWGQbaa beaakiabgsMiJkaadshaaiaawIcacaGLPaaaaSqaaiaadQgacqGHii IZcaWGZbaabeqdcqGHris5aOWaamWaaeaadaaeqbqaaiqadEhagaac amaaBaaaleaacaWGPbGaaGilaiaadQgaaeqaaaqaaiaadMgacqGHji YZcaWGZbaabeqdcqGHris5aOGaey4kaSYaaeWaaeaacqaHapaCdaqh aaWcbaGaamOAaaqaaiabgkHiTiaaigdaaaGccqGHsislcaaIXaaaca GLOaGaayzkaaGaeyOeI0YaaabuaeaaceWG3bGbaGaadaWgaaWcbaGa amyAaiaaiYcacaWGQbaabeaaaeaacaWGPbGaeyicI4Saam4Caaqab0 GaeyyeIuoakmaabmaabaGaeqiWda3aa0baaSqaaiaadMgaaeaacqGH sislcaaIXaaaaOGaeyOeI0IaaGymaaGaayjkaiaawMcaaaGaay5wai aaw2faaaGaayjkaiaawMcaaaqaaaqaaiaai2dadaWcaaqaaiaaigda aeaacaWGobWaaWbaaSqabeaacaaIYaaaaaaakmaaqafabeWcbaGaam OAaiabgIGiolaadohaaeqaniabggHiLdGcdaWadaqaaiaadEeadaqa daqaamaaeiaabaGaamiDaiabgkHiTiaad2gadaWgaaWcbaGaamOAaa qabaaakiaawIa7aiaaysW7caWG4bWaaSbaaSqaaiaadQgaaeqaaaGc caGLOaGaayzkaaGaeyOeI0Iaam4ramaaCaaaleqabaGaaGOmaaaakm aabmaabaWaaqGaaeaacaWG0bGaeyOeI0IaamyBamaaBaaaleaacaWG QbaabeaaaOGaayjcSdGaaGjbVlaadIhadaWgaaWcbaGaamOAaaqaba aakiaawIcacaGLPaaaaiaawUfacaGLDbaadaWadaqaamaaqafabaGa bm4DayaaiaWaaSbaaSqaaiaadMgacaaISaGaamOAaaqabaaabaGaam yAaiabgMGiplaadohaaeqaniabggHiLdGccqGHRaWkdaqadaqaaiab ec8aWnaaDaaaleaacaWGQbaabaGaeyOeI0IaaGymaaaakiabgkHiTi aaigdaaiaawIcacaGLPaaacqGHsisldaaeqbqaaiqadEhagaacamaa BaaaleaacaWGPbGaaGilaiaadQgaaeqaaaqaaiaadMgacqGHiiIZca WGZbaabeqdcqGHris5aOWaaeWaaeaacqaHapaCdaqhaaWcbaGaamyA aaqaaiabgkHiTiaaigdaaaGccqGHsislcaaIXaaacaGLOaGaayzkaa aacaGLBbGaayzxaaWaaWbaaSqabeaacaaIYaaaaaGcbaaabaGaaGyp amaalaaabaGaaGymaaqaaiaad6eadaahaaWcbeqaaiaaikdaaaaaaO WaaabuaeqaleaacaWGQbGaeyicI4Saam4Caaqab0GaeyyeIuoakmaa dmaabaGaam4ramaabmaabaWaaqGaaeaacaWG0bGaeyOeI0IaamyBam aaBaaaleaacaWGQbaabeaaaOGaayjcSdGaaGjbVlaadIhadaWgaaWc baGaamOAaaqabaaakiaawIcacaGLPaaacqGHsislcaWGhbWaaWbaaS qabeaacaaIYaaaaOWaaeWaaeaadaabcaqaaiaadshacqGHsislcaWG TbWaaSbaaSqaaiaadQgaaeqaaaGccaGLiWoacaaMe8UaamiEamaaBa aaleaacaWGQbaabeaaaOGaayjkaiaawMcaaaGaay5waiaaw2faamaa dmaabaWaaabuaeaaceWG3bGbaGaadaWgaaWcbaGaamyAaiaaiYcaca WGQbaabeaaaeaacaWGPbGaeyycI8Saam4Caaqab0GaeyyeIuoakiab gUcaRmaabmaabaGaeqiWda3aa0baaSqaaiaadQgaaeaacqGHsislca aIXaaaaOGaeyOeI0IaaGymaaGaayjkaiaawMcaamaabmaabaGaaGym aiabgkHiTmaaqafabaGabm4DayaaiaWaaSbaaSqaaiaadMgacaaISa GaamOAaaqabaaabaGaamyAaiabgIGiolaadohaaeqaniabggHiLdaa kiaawIcacaGLPaaaaiaawUfacaGLDbaadaahaaWcbeqaaiaaikdaaa GccqGHRaWkcaWGpbWaaeWaaeaacqaH7oaBcaWGUbWaaWbaaSqabeaa cqGHsislcaaIXaaaaaGccaGLOaGaayzkaaaabaaabaGaaGypamaala aabaGaaGymaaqaaiaad6gaaaWaaeWaaeaadaWcaaqaaiaad6eacqGH sislcaWGUbaabaGaamOtaaaaaiaawIcacaGLPaaadaahaaWcbeqaai aaikdaaaGcdaWdXaqabSqaaiaadggaaeaacaWGIbaaniabgUIiYdGc daWadaqaaiaadEeadaqadaqaamaaeiaabaGaamiDaiabgkHiTiaad2 gadaqadaqaaiaadIhaaiaawIcacaGLPaaacaaMc8oacaGLiWoacaaM e8UaamiEaaGaayjkaiaawMcaaiabgkHiTiaadEeadaahaaWcbeqaai aaikdaaaGcdaqadaqaamaaeiaabaGaamiDaiabgkHiTiaad2gadaqa daqaaiaadIhaaiaawIcacaGLPaaacaaMc8oacaGLiWoacaaMe8Uaam iEaaGaayjkaiaawMcaaaGaay5waiaaw2faamaadmaabaWaaSGbaeaa caWGObWaaSbaaSqaaiaaykW7ceWGZbGbaebaaeqaaOWaaeWaaeaaca WG4baacaGLOaGaayzkaaaabaGaamiAamaaBaaaleaacaWGZbaabeaa kmaabmaabaGaamiEaaGaayjkaiaawMcaaaaaaiaawUfacaGLDbaaca WGObWaaSbaaSqaaiaaykW7ceWGZbGbaebaaeqaaOWaaeWaaeaacaWG 4baacaGLOaGaayzkaaGaamizaiaadIhaaeaaaeaacaaMe8UaaGjbVl abgUcaRiaad+eadaqadaqaamaabmaabaGaamOBaiabeU7aSbGaayjk aiaawMcaamaaCaaaleqabaGaeyOeI0IaaGymaaaakiabeg7aHjabgU caRiabeU7aSjaad6gadaahaaWcbeqaaiabgkHiTiaaigdaaaaakiaa wIcacaGLPaaaaeaaaeaacaaI9aGaamyqamaaBaaaleaacaaIXaaabe aakiabgUcaRiaad+eadaqadaqaamaabmaabaGaamOBaiabeU7aSbGa ayjkaiaawMcaamaaCaaaleqabaGaeyOeI0IaaGymaaaakiabeg7aHj abgUcaRiabeU7aSjaad6gadaahaaWcbeqaaiabgkHiTiaaigdaaaaa kiaawIcacaGLPaaacaaIUaaaaaaa@7014@

Thus,

var ( F ˜ ( t ) F N ( t ) ) = var ( F ^ ( t ) F N ( t ) ) + O ( ( n λ ) 1 α + λ n 1 ) . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeODaiaabg gacaqGYbWaaeWaaeaaceWGgbGbaGaadaqadaqaaiaadshaaiaawIca caGLPaaacqGHsislcaWGgbWaaSbaaSqaaiaad6eaaeqaaOWaaeWaae aacaWG0baacaGLOaGaayzkaaaacaGLOaGaayzkaaGaaGypaiaabAha caqGHbGaaeOCamaabmaabaGabmOrayaajaWaaeWaaeaacaWG0baaca GLOaGaayzkaaGaeyOeI0IaamOramaaBaaaleaacaWGobaabeaakmaa bmaabaGaamiDaaGaayjkaiaawMcaaaGaayjkaiaawMcaaiabgUcaRi aad+eadaqadaqaamaabmaabaGaamOBaiabeU7aSbGaayjkaiaawMca amaaCaaaleqabaGaeyOeI0IaaGymaaaakiabeg7aHjabgUcaRiabeU 7aSjaad6gadaahaaWcbeqaaiabgkHiTiaaigdaaaaakiaawIcacaGL PaaacaaIUaaaaa@60C7@

Bias of the model-based estimator with modified fitted values

Let m ^ ^ i := k s w i , k m k , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmyBayaajy aajaWaaSbaaSqaaiaadMgaaeqaaOGaaGOoaiaai2dadaaeqaqabSqa aiaadUgacqGHiiIZcaWGZbaabeqdcqGHris5aOGaam4DamaaBaaale aacaWGPbGaaGilaiaadUgaaeqaaOGaamyBamaaBaaaleaacaWGRbaa beaakiaacYcaaaa@44A7@ c i , j := 1 w j , j + w i , j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4yamaaBa aaleaacaWGPbGaaGilaiaadQgaaeqaaOGaaGOoaiaai2dacaaIXaGa eyOeI0Iaam4DamaaBaaaleaacaWGQbGaaGilaiaadQgaaeqaaOGaey 4kaSIaam4DamaaBaaaleaacaWGPbGaaGilaiaadQgaaeqaaaaa@4446@ and

d i , j := 1 c i , j [ ( 1 c i , j ) ( t m i ) + ( m ^ ^ j m j ) ( m ^ ^ i m i ) + k s , k j ( w j , k w i , k ) ε k ] . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamizamaaBa aaleaacaWGPbGaaGilaiaadQgaaeqaaOGaaGOoaiaai2dadaWcaaqa aiaaigdaaeaacaWGJbWaaSbaaSqaaiaadMgacaaISaGaamOAaaqaba aaaOWaamWaaeaadaqadaqaaiaaigdacqGHsislcaWGJbWaaSbaaSqa aiaadMgacaaISaGaamOAaaqabaaakiaawIcacaGLPaaadaqadaqaai aadshacqGHsislcaWGTbWaaSbaaSqaaiaadMgaaeqaaaGccaGLOaGa ayzkaaGaey4kaSYaaeWaaeaaceWGTbGbaKGbaKaadaWgaaWcbaGaam OAaaqabaGccqGHsislcaWGTbWaaSbaaSqaaiaadQgaaeqaaaGccaGL OaGaayzkaaGaeyOeI0YaaeWaaeaaceWGTbGbaKGbaKaadaWgaaWcba GaamyAaaqabaGccqGHsislcaWGTbWaaSbaaSqaaiaadMgaaeqaaaGc caGLOaGaayzkaaGaey4kaSYaaabuaeqaleaacaWGRbGaeyicI4Saam 4CaiaaiYcacaWGRbGaeyiyIKRaamOAaaqab0GaeyyeIuoakmaabmaa baGaam4DamaaBaaaleaacaWGQbGaaGilaiaadUgaaeqaaOGaeyOeI0 Iaam4DamaaBaaaleaacaWGPbGaaGilaiaadUgaaeqaaaGccaGLOaGa ayzkaaGaeqyTdu2aaSbaaSqaaiaadUgaaeqaaaGccaGLBbGaayzxaa GaaGOlaaaa@74C3@

Observe that w i , j = O i , j ( ( n λ ) 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4DamaaBa aaleaacaWGPbGaaGilaiaadQgaaeqaaOGaaGypaiaad+eadaWgaaWc baGaamyAaiaaiYcacaWGQbaabeaakmaabmaabaWaaeWaaeaacaWGUb Gaeq4UdWgacaGLOaGaayzkaaWaaWbaaSqabeaacqGHsislcaaIXaaa aaGccaGLOaGaayzkaaaaaa@44C0@ so that

y j m ^ j t m ^ i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEamaaBa aaleaacaWGQbaabeaakiabgkHiTiqad2gagaqcamaaBaaaleaacaWG QbaabeaakiabgsMiJkaadshacqGHsislceWGTbGbaKaadaWgaaWcba GaamyAaaqabaaaaa@3FEC@

is (asymptotically, as soon as c i , j > 0 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeGaaeaaca WGJbWaaSbaaSqaaiaadMgacaaISaGaamOAaaqabaGccaaI+aGaaGim aaGaayzkaaaaaa@3AFA@ equivalent to

ε j t m i + d i , j . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyTdu2aaS baaSqaaiaadQgaaeqaaOGaeyizImQaamiDaiabgkHiTiaad2gadaWg aaWcbaGaamyAaaqabaGccqGHRaWkcaWGKbWaaSbaaSqaaiaadMgaca aISaGaamOAaaqabaGccaaIUaaaaa@42C7@

Since d i , j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamizamaaBa aaleaacaWGPbGaaGilaiaadQgaaeqaaaaa@38A7@ does not depend on ε j , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyTdu2aaS baaSqaaiaadQgaaeqaaOGaaiilaaaa@387B@ it follows that

E ( I ( y j m ^ j t m ^ i ) ) = E ( I ( ε j t m i + d i , j ) ) = E ( E ( I ( ε j t m i + d i , j ) | ε k , k j ) ) = E ( G ( t m i + d i , j | x j ) ) . ( A .1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqbaeaabmGaaa qaaiaadweadaqadaqaaiaadMeadaqadaqaaiaadMhadaWgaaWcbaGa amOAaaqabaGccqGHsislceWGTbGbaKaadaWgaaWcbaGaamOAaaqaba GccqGHKjYOcaWG0bGaeyOeI0IabmyBayaajaWaaSbaaSqaaiaadMga aeqaaaGccaGLOaGaayzkaaaacaGLOaGaayzkaaaabaGaaGypaiaadw eadaqadaqaaiaadMeadaqadaqaaiabew7aLnaaBaaaleaacaWGQbaa beaakiabgsMiJkaadshacqGHsislcaWGTbWaaSbaaSqaaiaadMgaae qaaOGaey4kaSIaamizamaaBaaaleaacaWGPbGaaGilaiaadQgaaeqa aaGccaGLOaGaayzkaaaacaGLOaGaayzkaaaabaaabaGaaGypaiaadw eadaqadaqaaiaadweadaqadaqaamaaeiaabaGaamysamaabmaabaGa eqyTdu2aaSbaaSqaaiaadQgaaeqaaOGaeyizImQaamiDaiabgkHiTi aad2gadaWgaaWcbaGaamyAaaqabaGccqGHRaWkcaWGKbWaaSbaaSqa aiaadMgacaaISaGaamOAaaqabaaakiaawIcacaGLPaaacaaMc8oaca GLiWoacaaMc8UaeqyTdu2aaSbaaSqaaiaadUgaaeqaaOGaaGilaiaa dUgacqGHGjsUcaWGQbaacaGLOaGaayzkaaaacaGLOaGaayzkaaaaba aabaGaaGypaiaadweadaqadaqaaiaadEeadaqadaqaamaaeiaabaGa amiDaiabgkHiTiaad2gadaWgaaWcbaGaamyAaaqabaGccqGHRaWkca WGKbWaaSbaaSqaaiaadMgacaaISaGaamOAaaqabaaakiaawIa7aiaa ysW7caWG4bWaaSbaaSqaaiaadQgaaeqaaaGccaGLOaGaayzkaaaaca GLOaGaayzkaaGaaGOlaaaacaaMf8UaaGzbVlaaywW7caaMf8UaaGzb VlaacIcacaGGbbGaaiOlaiaaigdacaGGPaaaaa@972B@

Now, using the fact that

d i , j = ( 1 c i , j ) ( t m i ) + ( m ^ ^ j m j ) ( m ^ ^ i m i ) + k s , k j ( w j , k w i , k ) ε k + R ( d i , j ) , ( A .2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamizamaaBa aaleaacaWGPbGaaGilaiaadQgaaeqaaOGaaGypamaabmaabaGaaGym aiabgkHiTiaadogadaWgaaWcbaGaamyAaiaaiYcacaWGQbaabeaaaO GaayjkaiaawMcaamaabmaabaGaamiDaiabgkHiTiaad2gadaWgaaWc baGaamyAaaqabaaakiaawIcacaGLPaaacqGHRaWkdaqadaqaaiqad2 gagaqcgaqcamaaBaaaleaacaWGQbaabeaakiabgkHiTiaad2gadaWg aaWcbaGaamOAaaqabaaakiaawIcacaGLPaaacqGHsisldaqadaqaai qad2gagaqcgaqcamaaBaaaleaacaWGPbaabeaakiabgkHiTiaad2ga daWgaaWcbaGaamyAaaqabaaakiaawIcacaGLPaaacqGHRaWkdaaeqb qabSqaaiaadUgacqGHiiIZcaWGZbGaaGilaiaadUgacqGHGjsUcaWG QbaabeqdcqGHris5aOWaaeWaaeaacaWG3bWaaSbaaSqaaiaadQgaca aISaGaam4AaaqabaGccqGHsislcaWG3bWaaSbaaSqaaiaadMgacaaI SaGaam4AaaqabaaakiaawIcacaGLPaaacqaH1oqzdaWgaaWcbaGaam 4AaaqabaGccqGHRaWkcaWGsbWaaeWaaeaacaWGKbWaaSbaaSqaaiaa dMgacaaISaGaamOAaaqabaaakiaawIcacaGLPaaacaaISaGaaGzbVl aaywW7caaMf8UaaiikaiaacgeacaGGUaGaaGOmaiaacMcaaaa@7CB9@

where

E 1 / 4 ( | R ( d i , j ) | 4 ) = O i , j ( λ n 1 + ( n λ ) 3 / 2 ) , ( A .3 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyramaaCa aaleqabaWaaSGbaeaacaaIXaaabaGaaGinaaaaaaGcdaqadaqaaiaa ykW7daabdaqaaiaadkfadaqadaqaaiaadsgadaWgaaWcbaGaamyAai aaiYcacaWGQbaabeaaaOGaayjkaiaawMcaaiaaykW7aiaawEa7caGL iWoadaahaaWcbeqaaiaaykW7caaI0aaaaaGccaGLOaGaayzkaaGaaG ypaiaad+eadaWgaaWcbaGaamyAaiaaiYcacaWGQbaabeaakmaabmaa baGaeq4UdWMaamOBamaaCaaaleqabaGaeyOeI0IaaGymaaaakiabgU caRmaabmaabaGaamOBaiabeU7aSbGaayjkaiaawMcaamaaCaaaleqa baGaeyOeI0YaaSGbaeaacaaIZaaabaGaaGOmaaaaaaaakiaawIcaca GLPaaacaaISaGaaGzbVlaaywW7caaMf8UaaGzbVlaaywW7caGGOaGa aiyqaiaac6cacaaIZaGaaiykaaaa@6622@

it is seen from (A.1) that

E ( I ( y j m ^ j t m ^ i ) ) = E ( G ( t m i + d i , j ) | x j ) = G ( t m i | x j ) + G ( 1,0 ) ( t m i | x j ) E ( d i , j ) + 1 2 G ( 2,0 ) ( t m i | x j ) E ( d i , j 2 ) + o i , j ( λ 4 + ( n λ ) 1 ) . ( A .4 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqbaeaabmGaaa qaaiaadweadaqadaqaaiaadMeadaqadaqaaiaadMhadaWgaaWcbaGa amOAaaqabaGccqGHsislceWGTbGbaKaadaWgaaWcbaGaamOAaaqaba GccqGHKjYOcaWG0bGaeyOeI0IabmyBayaajaWaaSbaaSqaaiaadMga aeqaaaGccaGLOaGaayzkaaaacaGLOaGaayzkaaaabaGaaGypaiaadw eadaqadaqaamaaeiaabaGaam4ramaabmaabaGaamiDaiabgkHiTiaa d2gadaWgaaWcbaGaamyAaaqabaGccqGHRaWkcaWGKbWaaSbaaSqaai aadMgacaaISaGaamOAaaqabaaakiaawIcacaGLPaaacaaMc8oacaGL iWoacaaMe8UaamiEamaaBaaaleaacaWGQbaabeaaaOGaayjkaiaawM caaaqaaaqaaiaai2dacaWGhbWaaeWaaeaadaabcaqaaiaadshacqGH sislcaWGTbWaaSbaaSqaaiaadMgaaeqaaaGccaGLiWoacaaMe8Uaam iEamaaBaaaleaacaWGQbaabeaaaOGaayjkaiaawMcaaiabgUcaRiaa dEeadaahaaWcbeqaamaabmaabaGaaGymaiaaiYcacaaIWaaacaGLOa GaayzkaaaaaOWaaeWaaeaadaabcaqaaiaadshacqGHsislcaWGTbWa aSbaaSqaaiaadMgaaeqaaaGccaGLiWoacaaMe8UaamiEamaaBaaale aacaWGQbaabeaaaOGaayjkaiaawMcaaiaadweadaqadaqaaiaadsga daWgaaWcbaGaamyAaiaaiYcacaWGQbaabeaaaOGaayjkaiaawMcaaa qaaaqaaiaaysW7caaMe8Uaey4kaSYaaSaaaeaacaaIXaaabaGaaGOm aaaacaWGhbWaaWbaaSqabeaadaqadaqaaiaaikdacaaISaGaaGimaa GaayjkaiaawMcaaaaakmaabmaabaWaaqGaaeaacaWG0bGaeyOeI0Ia amyBamaaBaaaleaacaWGPbaabeaaaOGaayjcSdGaaGjbVlaadIhada WgaaWcbaGaamOAaaqabaaakiaawIcacaGLPaaacaWGfbWaaeWaaeaa caWGKbWaa0baaSqaaiaadMgacaaISaGaamOAaaqaaiaaikdaaaaaki aawIcacaGLPaaacqGHRaWkcaWGVbWaaSbaaSqaaiaadMgacaaISaGa amOAaaqabaGcdaqadaqaaiabeU7aSnaaCaaaleqabaGaaGinaaaaki abgUcaRmaabmaabaGaamOBaiabeU7aSbGaayjkaiaawMcaamaaCaaa leqabaGaeyOeI0IaaGymaaaaaOGaayjkaiaawMcaaiaai6caaaGaaG zbVlaaywW7caaMf8UaaGzbVlaacIcacaGGbbGaaiOlaiaaisdacaGG Paaaaa@B24D@

Thus,

E ( F ^ * ( t ) F N ( t ) ) = E ( 1 N i s j s w i , j ( I ( y j m ^ j t m ^ i ) I ( y i t ) ) ) = 1 N i s j s w i , j [ G ( t m i | x j ) G ( t m i | x i ) ] + 1 N i s j s w i , j G ( 1 , 0 ) ( t m i | x j ) E ( d i , j ) + 1 2 N i s j s w i , j G ( 2 , 0 ) ( t m i | x j ) E ( d i , j 2 ) + o ( λ 4 + ( n λ ) 1 ) := C 1 + C 2 + C 3 + o ( λ 4 + ( n λ ) 1 ) . ( A .5 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqbaeaabuGaaa aabaGaamyramaabmaabaGabmOrayaajaWaaWbaaSqabeaacaaIQaaa aOGaaGzaVpaabmaabaGaamiDaaGaayjkaiaawMcaaiabgkHiTiaadA eadaWgaaWcbaGaamOtaaqabaGcdaqadaqaaiaadshaaiaawIcacaGL PaaaaiaawIcacaGLPaaaaeaacaaI9aGaamyramaabmaabaWaaSaaae aacaaIXaaabaGaamOtaaaadaaeqbqabSqaaiaadMgacqGHjiYZcaWG ZbaabeqdcqGHris5aOWaaabuaeaacaWG3bWaaSbaaSqaaiaadMgaca aISaGaamOAaaqabaaabaGaamOAaiabgIGiolaadohaaeqaniabggHi LdGcdaqadaqaaiaadMeadaqadaqaaiaadMhadaWgaaWcbaGaamOAaa qabaGccqGHsislceWGTbGbaKaadaWgaaWcbaGaamOAaaqabaGccqGH KjYOcaWG0bGaeyOeI0IabmyBayaajaWaaSbaaSqaaiaadMgaaeqaaa GccaGLOaGaayzkaaGaeyOeI0IaamysamaabmaabaGaamyEamaaBaaa leaacaWGPbaabeaakiabgsMiJkaadshaaiaawIcacaGLPaaaaiaawI cacaGLPaaaaiaawIcacaGLPaaaaeaaaeaacaaI9aWaaSaaaeaacaaI XaaabaGaamOtaaaadaaeqbqabSqaaiaadMgacqGHjiYZcaWGZbaabe qdcqGHris5aOWaaabuaeaacaWG3bWaaSbaaSqaaiaadMgacaaISaGa amOAaaqabaaabaGaamOAaiabgIGiolaadohaaeqaniabggHiLdGcda WadaqaaiaadEeadaqadaqaamaaeiaabaGaamiDaiabgkHiTiaad2ga daWgaaWcbaGaamyAaaqabaaakiaawIa7aiaaysW7caWG4bWaaSbaaS qaaiaadQgaaeqaaaGccaGLOaGaayzkaaGaeyOeI0Iaam4ramaabmaa baWaaqGaaeaacaWG0bGaeyOeI0IaamyBamaaBaaaleaacaWGPbaabe aaaOGaayjcSdGaaGjbVlaadIhadaWgaaWcbaGaamyAaaqabaaakiaa wIcacaGLPaaaaiaawUfacaGLDbaaaeaaaeaacaaMe8UaaGjbVlabgU caRmaalaaabaGaaGymaaqaaiaad6eaaaWaaabuaeqaleaacaWGPbGa eyycI8Saam4Caaqab0GaeyyeIuoakmaaqafabaGaam4DamaaBaaale aacaWGPbGaaGilaiaadQgaaeqaaOGaam4ramaaCaaaleqabaWaaeWa aeaacaaIXaGaaiilaiaaicdaaiaawIcacaGLPaaaaaaabaGaamOAai abgIGiolaadohaaeqaniabggHiLdGcdaqadaqaamaaeiaabaGaamiD aiabgkHiTiaad2gadaWgaaWcbaGaamyAaaqabaaakiaawIa7aiaays W7caWG4bWaaSbaaSqaaiaadQgaaeqaaaGccaGLOaGaayzkaaGaamyr amaabmaabaGaamizamaaBaaaleaacaWGPbGaaGilaiaadQgaaeqaaa GccaGLOaGaayzkaaaabaaabaGaaGjbVlaaysW7cqGHRaWkdaWcaaqa aiaaigdaaeaacaaIYaGaamOtaaaadaaeqbqabSqaaiaadMgacqGHji YZcaWGZbaabeqdcqGHris5aOWaaabuaeaacaWG3bWaaSbaaSqaaiaa dMgacaaISaGaamOAaaqabaGccaWGhbWaaWbaaSqabeaadaqadaqaai aaikdacaGGSaGaaGimaaGaayjkaiaawMcaaaaaaeaacaWGQbGaeyic I4Saam4Caaqab0GaeyyeIuoakmaabmaabaWaaqGaaeaacaWG0bGaey OeI0IaamyBamaaBaaaleaacaWGPbaabeaaaOGaayjcSdGaaGjbVlaa dIhadaWgaaWcbaGaamOAaaqabaaakiaawIcacaGLPaaacaWGfbWaae WaaeaacaWGKbWaa0baaSqaaiaadMgacaaISaGaamOAaaqaaiaaikda aaaakiaawIcacaGLPaaacqGHRaWkcaWGVbWaaeWaaeaacqaH7oaBda ahaaWcbeqaaiaaisdaaaGccqGHRaWkdaqadaqaaiaad6gacqaH7oaB aiaawIcacaGLPaaadaahaaWcbeqaaiabgkHiTiaaigdaaaaakiaawI cacaGLPaaaaeaaaeaacaaI6aGaaGypaiaadoeadaWgaaWcbaGaaGym aaqabaGccqGHRaWkcaWGdbWaaSbaaSqaaiaaikdaaeqaaOGaey4kaS Iaam4qamaaBaaaleaacaaIZaaabeaakiabgUcaRiaad+gadaqadaqa aiabeU7aSnaaCaaaleqabaGaaGinaaaakiabgUcaRmaabmaabaGaam OBaiabeU7aSbGaayjkaiaawMcaamaaCaaaleqabaGaeyOeI0IaaGym aaaaaOGaayjkaiaawMcaaiaai6caaaGaaGzbVlaaywW7caaMf8Uaai ikaiaacgeacaGGUaGaaGynaiaacMcaaaa@1927@

Consider first C 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4qamaaBa aaleaacaaIXaaabeaaaaa@36AE@ and note that

C 1 := 1 N i s j s w i , j [ G ( t m i | x j ) G ( t m i | x i ) ] = 1 2 N i s G ( 0 , 2 ) ( t m i | x i ) j s w i , j ( x j x i ) 2 + o ( λ 2 ) = λ 2 N n N μ 2 μ 0 a b G ( 0 , 2 ) ( t m ( x ) | x ) h s ¯ ( x ) d x + o ( λ 2 ) . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqbaeaabmGaaa qaaiaadoeadaWgaaWcbaGaaGymaaqabaaakeaacaaI6aGaaGypamaa laaabaGaaGymaaqaaiaad6eaaaWaaabuaeqaleaacaWGPbGaeyycI8 Saam4Caaqab0GaeyyeIuoakmaaqafabaGaam4DamaaBaaaleaacaWG PbGaaGilaiaadQgaaeqaaaqaaiaadQgacqGHiiIZcaWGZbaabeqdcq GHris5aOWaamWaaeaacaWGhbWaaeWaaeaadaabcaqaaiaadshacqGH sislcaWGTbWaaSbaaSqaaiaadMgaaeqaaaGccaGLiWoacaaMe8Uaam iEamaaBaaaleaacaWGQbaabeaaaOGaayjkaiaawMcaaiabgkHiTiaa dEeadaqadaqaamaaeiaabaGaamiDaiabgkHiTiaad2gadaWgaaWcba GaamyAaaqabaaakiaawIa7aiaaysW7caWG4bWaaSbaaSqaaiaadMga aeqaaaGccaGLOaGaayzkaaaacaGLBbGaayzxaaaabaaabaGaaGypam aalaaabaGaaGymaaqaaiaaikdacaWGobaaamaaqafabaGaam4ramaa CaaaleqabaWaaeWaaeaacaaIWaGaaiilaiaaikdaaiaawIcacaGLPa aaaaaabaGaamyAaiabgMGiplaadohaaeqaniabggHiLdGcdaqadaqa amaaeiaabaGaamiDaiabgkHiTiaad2gadaWgaaWcbaGaamyAaaqaba aakiaawIa7aiaaysW7caWG4bWaaSbaaSqaaiaadMgaaeqaaaGccaGL OaGaayzkaaWaaabuaeaacaWG3bWaaSbaaSqaaiaadMgacaaISaGaam OAaaqabaaabaGaamOAaiabgIGiolaadohaaeqaniabggHiLdGcdaqa daqaaiaadIhadaWgaaWcbaGaamOAaaqabaGccqGHsislcaWG4bWaaS baaSqaaiaadMgaaeqaaaGccaGLOaGaayzkaaWaaWbaaSqabeaacaaI YaaaaOGaey4kaSIaam4BamaabmaabaGaeq4UdW2aaWbaaSqabeaaca aIYaaaaaGccaGLOaGaayzkaaaabaaabaGaaGypaiabeU7aSnaaCaaa leqabaGaaGOmaaaakmaalaaabaGaamOtaiabgkHiTiaad6gaaeaaca WGobaaamaalaaabaGaeqiVd02aaSbaaSqaaiaaikdaaeqaaaGcbaGa eqiVd02aaSbaaSqaaiaaicdaaeqaaaaakmaapedabaGaam4ramaaCa aaleqabaWaaeWaaeaacaaIWaGaaiilaiaaikdaaiaawIcacaGLPaaa aaaabaGaamyyaaqaaiaadkgaa0Gaey4kIipakmaabmaabaWaaqGaae aacaWG0bGaeyOeI0IaamyBamaabmaabaGaamiEaaGaayjkaiaawMca aiaaykW7aiaawIa7aiaaysW7caWG4baacaGLOaGaayzkaaGaamiAam aaBaaaleaacaaMc8Uabm4CayaaraaabeaakmaabmaabaGaamiEaaGa ayjkaiaawMcaaiaadsgacaWG4bGaey4kaSIaam4BamaabmaabaGaeq 4UdW2aaWbaaSqabeaacaaIYaaaaaGccaGLOaGaayzkaaGaaGOlaaaa aaa@C271@

Consider next C 2 . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4qamaaBa aaleaacaaIYaaabeaakiaac6caaaa@376B@ (A.2) and (A.3) imply that

E ( d i , j ) = ( 1 c i , j ) ( t m i ) + ( m ^ ^ j m j ) ( m ^ ^ i m i ) + O i , j ( λ n 1 + ( n λ ) 3 / 2 ) = ( w j , j w i , j ) ( t m i ) + m j ′′ k s w j , k ( x k x j ) 2 m i ′′ k s w i , k ( x k x i ) 2 + o i , j ( λ 2 ) + O i , j ( λ n 1 + ( n λ ) 3 / 2 ) = ( w j , j w i , j ) ( t m i ) + ( m j ′′ m i ′′ ) k s w j , k ( x k x j ) 2 + m i ′′ ( k s w j , k ( x k x j ) 2 k s w i , k ( x k x i ) 2 ) + o i , j ( λ 2 ) + O i , j ( λ n 1 + ( n λ ) 3 / 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peee0hXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaafaqaaeGbca aaaeaacaWGfbWaaeWaaeaacaWGKbWaaSbaaSqaaiaadMgacaaISaGa amOAaaqabaaakiaawIcacaGLPaaaaeaacaaI9aWaaeWaaeaacaaIXa GaeyOeI0Iaam4yamaaBaaaleaacaWGPbGaaGilaiaadQgaaeqaaaGc caGLOaGaayzkaaWaaeWaaeaacaWG0bGaeyOeI0IaamyBamaaBaaale aacaWGPbaabeaaaOGaayjkaiaawMcaaiabgUcaRmaabmaabaGabmyB ayaajyaajaWaaSbaaSqaaiaadQgaaeqaaOGaeyOeI0IaamyBamaaBa aaleaacaWGQbaabeaaaOGaayjkaiaawMcaaiabgkHiTmaabmaabaGa bmyBayaajyaajaWaaSbaaSqaaiaadMgaaeqaaOGaeyOeI0IaamyBam aaBaaaleaacaWGPbaabeaaaOGaayjkaiaawMcaaiabgUcaRiaad+ea daWgaaWcbaGaamyAaiaaiYcacaWGQbaabeaakmaabmaabaGaeq4UdW MaamOBamaaCaaaleqabaGaeyOeI0IaaGymaaaakiabgUcaRmaabmaa baGaamOBaiabeU7aSbGaayjkaiaawMcaamaaCaaaleqabaGaeyOeI0 YaaSGbaeaacaaIZaaabaGaaGOmaaaaaaaakiaawIcacaGLPaaaaeaa aeaacaaI9aWaaeWaaeaacaWG3bWaaSbaaSqaaiaadQgacaaISaGaam OAaaqabaGccqGHsislcaWG3bWaaSbaaSqaaiaadMgacaaISaGaamOA aaqabaaakiaawIcacaGLPaaadaqadaqaaiaadshacqGHsislcaWGTb WaaSbaaSqaaiaadMgaaeqaaaGccaGLOaGaayzkaaGaey4kaSIabmyB ayaagaWaaSbaaSqaaiaadQgaaeqaaOGaaGjbVpaaqafabaGaam4Dam aaBaaaleaacaWGQbGaaGilaiaadUgaaeqaaOWaaeWaaeaacaWG4bWa aSbaaSqaaiaadUgaaeqaaOGaeyOeI0IaamiEamaaBaaaleaacaWGQb aabeaaaOGaayjkaiaawMcaamaaCaaaleqabaGaaGOmaaaaaeaacaWG RbGaeyicI4Saam4Caaqab0GaeyyeIuoakiabgkHiTiqad2gagaGbam aaBaaaleaacaWGPbaabeaakiaaysW7daaeqbqaaiaadEhadaWgaaWc baGaamyAaiaaiYcacaWGRbaabeaakmaabmaabaGaamiEamaaBaaale aacaWGRbaabeaakiabgkHiTiaadIhadaWgaaWcbaGaamyAaaqabaaa kiaawIcacaGLPaaadaahaaWcbeqaaiaaikdaaaaabaGaam4AaiabgI GiolaadohaaeqaniabggHiLdaakeaaaeaacaaMe8UaaGjbVlabgUca Riaad+gadaWgaaWcbaGaamyAaiaaiYcacaWGQbaabeaakmaabmaaba Gaeq4UdW2aaWbaaSqabeaacaaIYaaaaaGccaGLOaGaayzkaaGaey4k aSIaam4tamaaBaaaleaacaWGPbGaaGilaiaadQgaaeqaaOWaaeWaae aacqaH7oaBcaWGUbWaaWbaaSqabeaacqGHsislcaaIXaaaaOGaey4k aSYaaeWaaeaacaWGUbGaeq4UdWgacaGLOaGaayzkaaWaaWbaaSqabe aacqGHsisldaWcgaqaaiaaiodaaeaacaaIYaaaaaaaaOGaayjkaiaa wMcaaaqaaaqaaiaai2dadaqadaqaaiaadEhadaWgaaWcbaGaamOAai aaiYcacaWGQbaabeaakiabgkHiTiaadEhadaWgaaWcbaGaamyAaiaa iYcacaWGQbaabeaaaOGaayjkaiaawMcaamaabmaabaGaamiDaiabgk HiTiaad2gadaWgaaWcbaGaamyAaaqabaaakiaawIcacaGLPaaacqGH RaWkdaqadaqaaiqad2gagaGbamaaBaaaleaacaWGQbaabeaakiabgk HiTiqad2gagaGbamaaBaaaleaacaWGPbaabeaaaOGaayjkaiaawMca amaaqafabaGaam4DamaaBaaaleaacaWGQbGaaGilaiaadUgaaeqaaO WaaeWaaeaacaWG4bWaaSbaaSqaaiaadUgaaeqaaOGaeyOeI0IaamiE amaaBaaaleaacaWGQbaabeaaaOGaayjkaiaawMcaamaaCaaaleqaba GaaGOmaaaaaeaacaWGRbGaeyicI4Saam4Caaqab0GaeyyeIuoaaOqa aaqaaiaaysW7caaMe8Uaey4kaSIabmyBayaagaWaaSbaaSqaaiaadM gaaeqaaOGaaGjbVpaabmaabaWaaabuaeaacaWG3bWaaSbaaSqaaiaa dQgacaaISaGaam4AaaqabaGcdaqadaqaaiaadIhadaWgaaWcbaGaam 4AaaqabaGccqGHsislcaWG4bWaaSbaaSqaaiaadQgaaeqaaaGccaGL OaGaayzkaaWaaWbaaSqabeaacaaIYaaaaaqaaiaadUgacqGHiiIZca WGZbaabeqdcqGHris5aOGaeyOeI0YaaabuaeaacaWG3bWaaSbaaSqa aiaadMgacaaISaGaam4AaaqabaGcdaqadaqaaiaadIhadaWgaaWcba Gaam4AaaqabaGccqGHsislcaWG4bWaaSbaaSqaaiaadMgaaeqaaaGc caGLOaGaayzkaaWaaWbaaSqabeaacaaIYaaaaaqaaiaadUgacqGHii IZcaWGZbaabeqdcqGHris5aaGccaGLOaGaayzkaaaabaaabaGaaGjb VlaaysW7cqGHRaWkcaWGVbWaaSbaaSqaaiaadMgacaaISaGaamOAaa qabaGcdaqadaqaaiabeU7aSnaaCaaaleqabaGaaGOmaaaaaOGaayjk aiaawMcaaiabgUcaRiaad+eadaWgaaWcbaGaamyAaiaaiYcacaWGQb aabeaakmaabmaabaGaeq4UdWMaamOBamaaCaaaleqabaGaeyOeI0Ia aGymaaaakiabgUcaRmaabmaabaGaamOBaiabeU7aSbGaayjkaiaawM caamaaCaaaleqabaGaeyOeI0YaaSGbaeaacaaIZaaabaGaaGOmaaaa aaaakiaawIcacaGLPaaaaaaaaa@39BD@

so that

C 2 = C 2, a + C 2, b + C 2, c + o ( λ 2 ) + O ( λ n 1 + ( n λ ) 3 / 2 ) , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4qamaaBa aaleaacaaIYaaabeaakiaai2dacaWGdbWaaSbaaSqaaiaaikdacaaI SaGaamyyaaqabaGccqGHRaWkcaWGdbWaaSbaaSqaaiaaikdacaaISa GaamOyaaqabaGccqGHRaWkcaWGdbWaaSbaaSqaaiaaikdacaaISaGa am4yaaqabaGccqGHRaWkcaWGVbWaaeWaaeaacqaH7oaBdaahaaWcbe qaaiaaikdaaaaakiaawIcacaGLPaaacqGHRaWkcaWGpbWaaeWaaeaa cqaH7oaBcaWGUbWaaWbaaSqabeaacqGHsislcaaIXaaaaOGaey4kaS YaaeWaaeaacaWGUbGaeq4UdWgacaGLOaGaayzkaaWaaWbaaSqabeaa cqGHsisldaWcgaqaaiaaiodaaeaacaaIYaaaaaaaaOGaayjkaiaawM caaiaaiYcaaaa@598E@

where

C 2, a := 1 N i s j s w i , j G ( 1 , 0 ) ( t m i | x j ) ( w j , j w i , j ) ( t m i ) = 1 N i s G ( 1 , 0 ) ( t m i | x i ) ( t m i ) j s w i , j ( w j , j w i , j ) + O ( n 1 ) = 1 n λ N n N K ( 0 ) κ μ 0 a b G ( 1 , 0 ) ( t m ( x ) | x ) ( t m ( x ) ) [ h s ¯ ( x ) / h s ( x ) ] d x + O ( ( n λ ) 1 λ 1 α + n 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqbaeaabqGaaa aabaGaam4qamaaBaaaleaacaaIYaGaaGilaiaadggaaeqaaaGcbaGa aGOoaiaai2dadaWcaaqaaiaaigdaaeaacaWGobaaamaaqafabeWcba GaamyAaiabgMGiplaadohaaeqaniabggHiLdGcdaaeqbqaaiaadEha daWgaaWcbaGaamyAaiaaiYcacaWGQbaabeaakiaadEeadaahaaWcbe qaamaabmaabaGaaGymaiaacYcacaaIWaaacaGLOaGaayzkaaaaaaqa aiaadQgacqGHiiIZcaWGZbaabeqdcqGHris5aOWaaeWaaeaadaabca qaaiaadshacqGHsislcaWGTbWaaSbaaSqaaiaadMgaaeqaaaGccaGL iWoacaaMe8UaamiEamaaBaaaleaacaWGQbaabeaaaOGaayjkaiaawM caamaabmaabaGaam4DamaaBaaaleaacaWGQbGaaGilaiaadQgaaeqa aOGaeyOeI0Iaam4DamaaBaaaleaacaWGPbGaaGilaiaadQgaaeqaaa GccaGLOaGaayzkaaWaaeWaaeaacaWG0bGaeyOeI0IaamyBamaaBaaa leaacaWGPbaabeaaaOGaayjkaiaawMcaaaqaaaqaaiaai2dadaWcaa qaaiaaigdaaeaacaWGobaaamaaqafabaGaam4ramaaCaaaleqabaWa aeWaaeaacaaIXaGaaiilaiaaicdaaiaawIcacaGLPaaaaaaabaGaam yAaiabgMGiplaadohaaeqaniabggHiLdGcdaqadaqaamaaeiaabaGa amiDaiabgkHiTiaad2gadaWgaaWcbaGaamyAaaqabaaakiaawIa7ai aaysW7caWG4bWaaSbaaSqaaiaadMgaaeqaaaGccaGLOaGaayzkaaWa aeWaaeaacaWG0bGaeyOeI0IaamyBamaaBaaaleaacaWGPbaabeaaaO GaayjkaiaawMcaamaaqafabaGaam4DamaaBaaaleaacaWGPbGaaGil aiaadQgaaeqaaaqaaiaadQgacqGHiiIZcaWGZbaabeqdcqGHris5aO WaaeWaaeaacaWG3bWaaSbaaSqaaiaadQgacaaISaGaamOAaaqabaGc cqGHsislcaWG3bWaaSbaaSqaaiaadMgacaaISaGaamOAaaqabaaaki aawIcacaGLPaaacqGHRaWkcaWGpbWaaeWaaeaacaWGUbWaaWbaaSqa beaacqGHsislcaaIXaaaaaGccaGLOaGaayzkaaaabaaabaGaaGypam aalaaabaGaaGymaaqaaiaad6gacqaH7oaBaaWaaSaaaeaacaWGobGa eyOeI0IaamOBaaqaaiaad6eaaaWaaSaaaeaacaWGlbWaaeWaaeaaca aIWaaacaGLOaGaayzkaaGaeyOeI0IaeqOUdSgabaGaeqiVd02aaSba aSqaaiaaicdaaeqaaaaakmaapedabeWcbaGaamyyaaqaaiaadkgaa0 Gaey4kIipakiaadEeadaahaaWcbeqaamaabmaabaGaaGymaiaacYca caaIWaaacaGLOaGaayzkaaaaaOWaaeWaaeaadaabcaqaaiaadshacq GHsislcaWGTbWaaeWaaeaacaWG4baacaGLOaGaayzkaaGaaGPaVdGa ayjcSdGaaGjbVlaadIhaaiaawIcacaGLPaaadaqadaqaaiaadshacq GHsislcaWGTbWaaeWaaeaacaWG4baacaGLOaGaayzkaaaacaGLOaGa ayzkaaWaamWaaeaadaWcgaqaaiaadIgadaWgaaWcbaGaaGPaVlqado hagaqeaaqabaGcdaqadaqaaiaadIhaaiaawIcacaGLPaaaaeaacaWG ObWaaSbaaSqaaiaadohaaeqaaOWaaeWaaeaacaWG4baacaGLOaGaay zkaaaaaaGaay5waiaaw2faaiaadsgacaWG4baabaaabaGaaGjbVlaa ysW7cqGHRaWkcaWGpbWaaeWaaeaadaqadaqaaiaad6gacqaH7oaBai aawIcacaGLPaaadaahaaWcbeqaaiabgkHiTiaaigdaaaGccqaH7oaB daahaaWcbeqaaiabgkHiTiaaigdaaaGccqaHXoqycqGHRaWkcaWGUb WaaWbaaSqabeaacqGHsislcaaIXaaaaaGccaGLOaGaayzkaaaaaaaa @F03A@

with κ := 1 1 K 2 ( u ) d u , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOUdSMaaG Ooaiaai2dadaWdXaqaaiaadUeadaahaaWcbeqaaiaaikdaaaaabaGa eyOeI0IaaGymaaqaaiaaigdaa0Gaey4kIipakmaabmaabaGaamyDaa GaayjkaiaawMcaaiaadsgacaWG1bGaaiilaaaa@4396@

C 2, b := 1 N i s j s w i , j G ( 1 , 0 ) ( t m i | x j ) ( m j ′′ m i ′′ ) k s w j , k ( x k x j ) 2 = o (