A short note on quantile and expectile estimation in unequal probability samples 5. Simulations

We run a small simulation study to show the performance of the expectile based estimates. In the following, we make use of the Mizuno sampling method (see Midzuno 1952) and define the inclusion probabilities π j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqWqpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaeqiWda3aaS baaSqaaiaadQgaaeqaaaaa@39E4@  proportional to a measure of size x , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqWqpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamiEaiaacY caaaa@38B9@  see R package “sampling” by Tillé and Matei (2015). We examine two data sets also used in Kuk (1988). The first data set (Dwellings) contains two variables, the number of dwelling units ( X ) , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqWqpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaWaaeWaaeaaca WGybaacaGLOaGaayzkaaGaaiilaaaa@3A22@  and the number of rented units ( Y ) , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqWqpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaWaaeWaaeaaca WGzbaacaGLOaGaayzkaaGaaiilaaaa@3A23@  which are highly correlated (with a correlation of 0.97); see also Kish (1965). The second data set (Villages) includes information on the population ( X ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqWqpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaWaaeWaaeaaca WGybaacaGLOaGaayzkaaaaaa@3972@  and on the number of workers in household industry ( Y ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqWqpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaWaaeWaaeaaca WGzbaacaGLOaGaayzkaaaaaa@3973@  for 128 villages in India; see Murthy (1967). In the second data set the correlation between Y MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqWqpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamywaaaa@37EA@  and X MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqWqpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamiwaaaa@37E9@  is 0.54. In order to compare our simulation results with the results of Kuk (1988) we choose the same sample size of n = 30 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqWqpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamOBaiaai2 dacaaIZaGaaGimaaaa@3A3D@  (from a total population of N = 270 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqWqpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamOtaiaai2 dacaaIYaGaaG4naiaaicdaaaa@3ADD@  for the Dwellings data and N = 128 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqWqpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamOtaiaai2 dacaaIXaGaaGOmaiaaiIdaaaa@3ADF@  for the Villages data).

We compare quantiles defined by inversion of F ^ R MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqWqpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGabmOrayaaja WaaSbaaSqaaiaadkfaaeqaaaaa@38EA@ with quantiles defined by inversion of F ^ R M . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqWqpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGabmOrayaaja Waa0baaSqaaiaadkfaaeaacaWGnbaaaOGaaiOlaaaa@3A79@ In Table 5.1 we give the root mean squared error (RMSE) and the relative efficiency for specified quantiles. We note that the median for the village data and for the Dwelling data also upper quantiles derived from expectiles yield increased efficiency. Also the efficiency gain does not hold uniformly as we observe a loss of efficiency for lower quantiles.

Table 5.1
Comparison of mean squared error on a basis of 500 replications
Table summary
This table displays the results of Comparison of mean squared error on a basis of 500 replications XXXX, quantiles XXXX, quantiles from expectiles XXXX and relative efficiency XXXX (appearing as column headers).
  α MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqWqpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbeqabeWaceGabiqabeqabmqabeabbaGcbaGaeqySdegaaa@3AD8@ quantiles
MSE ( Q ^ R ( α ) ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqWqpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbeqabeWaceGabiqabeqabmqabeabbaGcbaWaaOaaaeaaca qGnbGaae4uaiaabweadaqadaqaaiqadgfagaqcamaaBaaaleaacaWG sbaabeaakmaabmaabaGaeqySdegacaGLOaGaayzkaaaacaGLOaGaay zkaaaaleqaaaaa@4266@
quantiles from expectiles
MSE ( Q ^ R M ( α ) ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqWqpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbeqabeWaceGabiqabeqabmqabeabbaGcbaWaaOaaaeaaca qGnbGaae4uaiaabweadaqadaqaaiqadgfagaqcamaaDaaaleaacaWG sbaabaGaamytaaaakmaabmaabaGaeqySdegacaGLOaGaayzkaaaaca GLOaGaayzkaaaaleqaaaaa@4339@
relative efficiency
MSE ( Q ^ R M ( α ) ) MSE ( Q ^ R ( α ) ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqWqpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbeqabeWaceGabiqabeqabmqabeabbaGcbaWaaSaaaeaada Gcaaqaaiaab2eacaqGtbGaaeyramaabmaabaGabmyuayaajaWaa0ba aSqaaiaadkfaaeaacaWGnbaaaOWaaeWaaeaacqaHXoqyaiaawIcaca GLPaaaaiaawIcacaGLPaaaaSqabaaakeaadaGcaaqaaiaab2eacaqG tbGaaeyramaabmaabaGabmyuayaajaWaaSbaaSqaaiaadkfaaeqaaO WaaeWaaeaacqaHXoqyaiaawIcacaGLPaaaaiaawIcacaGLPaaaaSqa baaaaaaa@4C80@
Dwellings 0.1 2.57 2.76 1.07
0.25 1.77 1.97 1.11
0.5 2.45 2.35 0.96
0.75 3.15 2.91 0.92
0.9 4.20 3.43 0.82
Villages 0.1 5.52 6.65 1.21
0.25 11.41 10.31 0.90
0.5 12.29 11.69 0.95
0.75 16.24 15.41 0.95
0.9 13.31 18.34 1.38

To obtain more insight we run a simulation scenario which involves a larger sample size of n = 100 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqWqpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamOBaiaai2 dacaaIXaGaaGimaiaaicdaaaa@3AF5@ selected from populations of sizes N = 1,000 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqWqpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamOtaiaai2 dacaqGXaGaaeilaiaabcdacaqGWaGaaeimaaaa@3C22@ and N = 10,000 . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqWqpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamOtaiaai2 dacaqGXaGaaeimaiaabYcacaqGWaGaaeimaiaabcdacaqGUaaaaa@3D86@ We draw Y MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqWqpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamywaaaa@37EA@ and X MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqWqpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamiwaaaa@37E9@ from a bivariate log standard normal distribution with μ = 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqWqpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaeqiVd0MaaG ypaiaaicdaaaa@3A43@ and σ = 1. MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqWqpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaeq4WdmNaaG ypaiaaigdacaGGUaaaaa@3B03@ The variables Y MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqWqpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamywaaaa@37EA@ and X MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqWqpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamiwaaaa@37E9@ are drawn such that the correlation between the variables is equal to 0.9. We again calculate the root mean squared error for a range of α MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqWqpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaeqySdegaaa@38AB@ values and show the relative efficiency of the expectile based approach in Figure 5.1. For better visual presentation we show a smoothed version of the relative efficiency. We notice a reduction in the root mean squared error for both cases N = 1,000 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqWqpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamOtaiaai2 dacaqGXaGaaeilaiaabcdacaqGWaGaaeimaaaa@3C22@ and N = 10,000 . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqWqpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamOtaiaai2 dacaqGXaGaaeimaiaabYcacaqGWaGaaeimaiaabcdacaGGUaaaaa@3D87@ We may conclude that the expectiles can easily be fitted in unequal probability sampling and the relation between expectiles and the distribution function can be used numerically to calculate quantiles with increased efficiency. This efficiency gain holds for upper quantiles only, that is for α MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqWqpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaeqySdegaaa@38AB@ bounded away from zero. Note however that the sampling scheme is such that large values of Y MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqWqpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamywaaaa@37EA@ are sampled with higher probability, reflecting that the sampling scheme aims to get more reliable estimates for the right hand side of the distribution function, i.e., for large quantiles. If we are interested in small quantiles we should use a different samling scheme by giving individuals with small values of Y MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqWqpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamywaaaa@37EA@ an increased inclusion probability. In this case the behavior shown in Figure 5.1 would be mirrored with respect to α . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqWqpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaeqySdeMaai Olaaaa@395D@

Figure 5.1 of article 14545

Description of Figure 5.1

Figure made of two graphs presenting the relative root mean squared error of quantiles and quantiles from expectiles for the Probability Proportional to Size (PPS) design calculated from 500 repetitions, for N=1,000 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGobGaaG ypaiaabgdacaqGSaGaaeimaiaabcdacaqGWaaaaa@3D76@  and N=10,000. MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGobGaaG ypaiaabgdacaqGWaGaaeilaiaabcdacaqGWaGaaeimaiaac6caaaa@3EDB@  For both graphs, the y axis is the ratio of RMSE quantiles from F R M MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGgbWaa0 baaSqaaiaadkfaaeaacaWGnbaaaaaa@3B01@  and from F R , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGgbWaaS baaSqaaiaadkfaaeqaaOGaaiilaaaa@3AE8@  going from 0.90 to 1.15. α MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHXoqyaa a@39FF@  is on the x axis, going from 0.01 to 0.99. For N=1,000, MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGobGaaG ypaiaabgdacaqGSaGaaeimaiaabcdacaqGWaGaaeilaaaa@3E25@  the ratio is close to 1.15 for small α MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHXoqycq GHsislaaa@3AEC@  values before decreasing between 0.90 and 0.95 for an α MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHXoqycq GHsislaaa@3AEC@  value of about 0.25. After, the ratio is globally increasing slowly toward 1.00 when α MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHXoqyaa a@39FF@  increases. For N=10,000, MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGobGaaG ypaiaabgdacaqGWaGaaeilaiaabcdacaqGWaGaaeimaiaabYcaaaa@3ED8@  the ratio is close to 1.10 for small α MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHXoqycq GHsislaaa@3AEC@  values before decreasing to about 0.95 for α MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHXoqyaa a@39FF@  between 0.20 and 0.25. After, the ratio is globally increasing more quickly toward 1.00 when α MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHXoqyaa a@39FF@  increases.

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