A comparison between nonparametric estimators for finite population distribution functions 4. Design-based properties

In the previous section we have shown that the model-based estimators F ^ ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaja WaaeWaaeaacaWG0baacaGLOaGaayzkaaaaaa@385C@ and F ^ * ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaja WaaWbaaSqabeaacaaIQaaaaOGaaGzaVpaabmaabaGaamiDaaGaayjk aiaawMcaaaaa@3AD1@ are asymptotically model-unbiased and model mean square error consistent. However, they are not design-unbiased in general and therefore they should not be used when the sample inclusion probabilities are not constant. In these cases the generalized difference estimators F ˜ ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaia WaaeWaaeaacaWG0baacaGLOaGaayzkaaaaaa@385B@ and F ˜ * ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaia WaaWbaaSqabeaacaaIQaaaaOGaaGzaVpaabmaabaGaamiDaaGaayjk aiaawMcaaaaa@3AD0@ should be used. In fact, it follows from the results in Breidt and Opsomer (2000) that under fairly general conditions F ˜ ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaia WaaeWaaeaacaWG0baacaGLOaGaayzkaaaaaa@385B@ is asymptotically design-unbiased and that its design mean square error is given by

E d ( | F ˜ ( t ) F N ( t ) | 2 ) = 1 N 2 i , j U π i , j π i π j π i π j [ I ( y i t ) G ¯ i ( t ) ] [ I ( y j t ) G ¯ j ( t ) ] + o ( n 1 ) , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyramaaBa aaleaacaWGKbaabeaakmaabmaabaWaaqWaaeaaceWGgbGbaGaadaqa daqaaiaadshaaiaawIcacaGLPaaacqGHsislcaWGgbWaaSbaaSqaai aad6eaaeqaaOWaaeWaaeaacaWG0baacaGLOaGaayzkaaGaaGPaVdGa ay5bSlaawIa7amaaCaaaleqabaGaaGOmaaaaaOGaayjkaiaawMcaai aai2dadaWcaaqaaiaaigdaaeaacaWGobWaaWbaaSqabeaacaaIYaaa aaaakmaaqafabeWcbaGaamyAaiaaiYcacaWGQbGaeyicI4Saamyvaa qab0GaeyyeIuoakmaalaaabaGaeqiWda3aaSbaaSqaaiaadMgacaaI SaGaamOAaaqabaGccqGHsislcqaHapaCdaWgaaWcbaGaamyAaaqaba GccqaHapaCdaWgaaWcbaGaamOAaaqabaaakeaacqaHapaCdaWgaaWc baGaamyAaaqabaGccqaHapaCdaWgaaWcbaGaamOAaaqabaaaaOWaam WaaeaacaWGjbWaaeWaaeaacaWG5bWaaSbaaSqaaiaadMgaaeqaaOGa eyizImQaamiDaaGaayjkaiaawMcaaiabgkHiTiqadEeagaqeamaaBa aaleaacaWGPbaabeaakmaabmaabaGaamiDaaGaayjkaiaawMcaaaGa ay5waiaaw2faamaadmaabaGaamysamaabmaabaGaamyEamaaBaaale aacaWGQbaabeaakiabgsMiJkaadshaaiaawIcacaGLPaaacqGHsisl ceWGhbGbaebadaWgaaWcbaGaamOAaaqabaGcdaqadaqaaiaadshaai aawIcacaGLPaaaaiaawUfacaGLDbaacqGHRaWkcaWGVbWaaeWaaeaa caWGUbWaaWbaaSqabeaacqGHsislcaaIXaaaaaGccaGLOaGaayzkaa GaaGilaaaa@8638@

where E d ( ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyramaaBa aaleaacaWGKbaabeaakmaabmaabaGaeyyXICnacaGLOaGaayzkaaaa aa@3ABB@ denotes expectation with respect to the sample design, π i , j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiWda3aaS baaSqaaiaadMgacaaISaGaamOAaaqabaaaaa@397B@ denotes the joint sample inclusion probability for units i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAaaaa@35ED@ and j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOAaaaa@35EE@ (it is understood that π i , i = π i ) , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeGaaeaacq aHapaCdaWgaaWcbaGaamyAaiaaiYcacaWGPbaabeaakiaai2dacqaH apaCdaWgaaWcbaGaamyAaaqabaaakiaawMcaaiaacYcaaaa@3EA4@ and where

G ¯ i ( t ) := j U w ¯ i , j I ( y j t ) . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabm4rayaara WaaSbaaSqaaiaadMgaaeqaaOWaaeWaaeaacaWG0baacaGLOaGaayzk aaGaaGOoaiaai2dadaaeqbqaaiqadEhagaqeamaaBaaaleaacaWGPb GaaGilaiaadQgaaeqaaOGaamysamaabmaabaGaamyEamaaBaaaleaa caWGQbaabeaakiabgsMiJkaadshaaiaawIcacaGLPaaaaSqaaiaadQ gacqGHiiIZcaWGvbaabeqdcqGHris5aOGaaGOlaaaa@4C4A@

The regression weights w ¯ i , j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabm4Dayaara WaaSbaaSqaaiaadMgacaaISaGaamOAaaqabaaaaa@38D2@ in the definition of G ¯ i ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabm4rayaara WaaSbaaSqaaiaadMgaaeqaaOWaaeWaaeaacaWG0baacaGLOaGaayzk aaaaaa@3989@ refer to the whole finite population U MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyvaaaa@35D9@ and are given by

w ¯ i , j  : = 1 N λ K ( x i x j λ ) M ¯ 2, s ( x i ) ( x i x j λ ) M ¯ 1, s ( x i ) M ¯ 2, s ( x i ) M ¯ 0, s ( x i ) M ¯ 1, s 2 ( x i ) , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabm4Dayaara WaaSbaaSqaaiaadMgacaaISaGaamOAaaqabaGccaaI6aGaaGypamaa laaabaGaaGymaaqaaiaad6eacqaH7oaBaaGaam4samaabmaabaWaaS aaaeaacaWG4bWaaSbaaSqaaiaadMgaaeqaaOGaeyOeI0IaamiEamaa BaaaleaacaWGQbaabeaaaOqaaiabeU7aSbaaaiaawIcacaGLPaaada Wcaaqaaiqad2eagaqeamaaBaaaleaacaaIYaGaaGilaiaadohaaeqa aOWaaeWaaeaacaWG4bWaaSbaaSqaaiaadMgaaeqaaaGccaGLOaGaay zkaaGaeyOeI0YaaeWaaeaadaWcaaqaaiaadIhadaWgaaWcbaGaamyA aaqabaGccqGHsislcaWG4bWaaSbaaSqaaiaadQgaaeqaaaGcbaGaeq 4UdWgaaaGaayjkaiaawMcaaiqad2eagaqeamaaBaaaleaacaaIXaGa aGilaiaadohaaeqaaOWaaeWaaeaacaWG4bWaaSbaaSqaaiaadMgaae qaaaGccaGLOaGaayzkaaaabaGabmytayaaraWaaSbaaSqaaiaaikda caaISaGaam4CaaqabaGcdaqadaqaaiaadIhadaWgaaWcbaGaamyAaa qabaaakiaawIcacaGLPaaaceWGnbGbaebadaWgaaWcbaGaaGimaiaa iYcacaWGZbaabeaakmaabmaabaGaamiEamaaBaaaleaacaWGPbaabe aaaOGaayjkaiaawMcaaiabgkHiTiqad2eagaqeamaaDaaaleaacaaI XaGaaGilaiaadohaaeaacaaIYaaaaOWaaeWaaeaacaWG4bWaaSbaaS qaaiaadMgaaeqaaaGccaGLOaGaayzkaaaaaiaaiYcaaaa@76DF@

where

M ¯ r , s ( x ) := k U 1 N λ K ( x x k λ ) ( x x k λ ) r , r = 0,1,2. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmytayaara WaaSbaaSqaaiaadkhacaaISaGaam4CaaqabaGcdaqadaqaaiaadIha aiaawIcacaGLPaaacaaI6aGaaGypamaaqafabeWcbaGaam4AaiabgI GiolaadwfaaeqaniabggHiLdGcdaWcaaqaaiaaigdaaeaacaWGobGa eq4UdWgaaiaadUeadaqadaqaamaalaaabaGaamiEaiabgkHiTiaadI hadaWgaaWcbaGaam4AaaqabaaakeaacqaH7oaBaaaacaGLOaGaayzk aaWaaeWaaeaadaWcaaqaaiaadIhacqGHsislcaWG4bWaaSbaaSqaai aadUgaaeqaaaGcbaGaeq4UdWgaaaGaayjkaiaawMcaamaaCaaaleqa baGaamOCaaaakiaaiYcacaaMf8UaaGzbVlaaywW7caWGYbGaaGypai aaicdacaaISaGaaGymaiaaiYcacaaIYaGaaGOlaaaa@61C7@

Moreover, according to Breidt and Opsomer (2000),

V ˜ ( F ˜ ( 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 ) G ˜ i ( t ) ] [ I ( y j t ) G ˜ j ( t ) ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOvayaaia WaaeWaaeaaceWGgbGbaGaadaqadaqaaiaadshaaiaawIcacaGLPaaa aiaawIcacaGLPaaacaaI6aGaaGypamaalaaabaGaaGymaaqaaiaad6 eadaahaaWcbeqaaiaaikdaaaaaaOWaaabuaeqaleaacaWGPbGaaGil aiaadQgacqGHiiIZcaWGZbaabeqdcqGHris5aOWaaSaaaeaacqaHap aCdaWgaaWcbaGaamyAaiaaiYcacaWGQbaabeaakiabgkHiTiabec8a WnaaBaaaleaacaWGPbaabeaakiabec8aWnaaBaaaleaacaWGQbaabe aaaOqaaiabec8aWnaaBaaaleaacaWGPbGaaGilaiaadQgaaeqaaOGa eqiWda3aaSbaaSqaaiaadMgaaeqaaOGaeqiWda3aaSbaaSqaaiaadQ gaaeqaaaaakmaadmaabaGaamysamaabmaabaGaamyEamaaBaaaleaa caWGPbaabeaakiabgsMiJkaadshaaiaawIcacaGLPaaacqGHsislce WGhbGbaGaadaWgaaWcbaGaamyAaaqabaGcdaqadaqaaiaadshaaiaa wIcacaGLPaaaaiaawUfacaGLDbaadaWadaqaaiaadMeadaqadaqaai aadMhadaWgaaWcbaGaamOAaaqabaGccqGHKjYOcaWG0baacaGLOaGa ayzkaaGaeyOeI0Iabm4rayaaiaWaaSbaaSqaaiaadQgaaeqaaOWaae WaaeaacaWG0baacaGLOaGaayzkaaaacaGLBbGaayzxaaaaaa@78C5@

is a consistent estimator for the design mean square error of F ˜ ( t ) . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaia WaaeWaaeaacaWG0baacaGLOaGaayzkaaGaaiOlaaaa@390D@

Unfortunately the results in Breidt and Opsomer (2000) cannot be applied to the generalized difference estimator F ˜ * ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaia WaaWbaaSqabeaacaaIQaaaaOGaaGzaVpaabmaabaGaamiDaaGaayjk aiaawMcaaaaa@3AD0@ as well, since the latter estimator does not fall into the class of local polynomial regression estimators due to the presence of the regression function estimators m ˜ i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmyBayaaia WaaSbaaSqaaiaadMgaaeqaaaaa@371A@ and m ˜ j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmyBayaaia WaaSbaaSqaaiaadQgaaeqaaaaa@371B@ inside the indicator functions in the fitted values G ˜ i * ( t ) . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabm4rayaaia Waa0baaSqaaiaadMgaaeaacaaIQaaaaOGaaGzaVpaabmaabaGaamiD aaGaayjkaiaawMcaaiaac6caaaa@3C71@ However, the results for F ˜ ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaia WaaeWaaeaacaWG0baacaGLOaGaayzkaaaaaa@385B@ suggest that in large samples G ˜ i * ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabm4rayaaia Waa0baaSqaaiaadMgaaeaacaaIQaaaaOWaaeWaaeaacaWG0baacaGL OaGaayzkaaaaaa@3A35@ and

G ¯ i * ( t ) := j U w ¯ i , j I ( y j m ¯ j t m ¯ i ) , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabm4rayaara Waa0baaSqaaiaadMgaaeaacaaIQaaaaOWaaeWaaeaacaWG0baacaGL OaGaayzkaaGaaGOoaiaai2dadaaeqbqaaiqadEhagaqeamaaBaaale aacaWGPbGaaGilaiaadQgaaeqaaOGaamysamaabmaabaGaamyEamaa BaaaleaacaWGQbaabeaakiabgkHiTiqad2gagaqeamaaBaaaleaaca WGQbaabeaakiabgsMiJkaadshacqGHsislceWGTbGbaebadaWgaaWc baGaamyAaaqabaaakiaawIcacaGLPaaaaSqaaiaadQgacqGHiiIZca WGvbaabeqdcqGHris5aOGaaGilaaaa@5334@

where m ¯ i := j U w ¯ i , j y j , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmyBayaara WaaSbaaSqaaiaadMgaaeqaaOGaaGOoaiaai2dadaaeqaqaaiqadEha gaqeamaaBaaaleaacaWGPbGaaGilaiaadQgaaeqaaOGaamyEamaaBa aaleaacaWGQbaabeaaaeaacaWGQbGaeyicI4Saamyvaaqab0Gaeyye IuoakiaacYcaaaa@448D@ are approximately the same, and that

E d ( | F ˜ * ( t ) F N ( t ) | 2 ) = 1 N 2 i , j U π i , j π i π j π i π j [ I ( y i t ) G ¯ i * ( t ) ] [ I ( y j t ) G ¯ j * ( t ) ] + o ( n 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyramaaBa aaleaacaWGKbaabeaakmaabmaabaWaaqWaaeaacaaMc8UabmOrayaa iaWaaWbaaSqabeaacaGGQaaaaOWaaeWaaeaacaWG0baacaGLOaGaay zkaaGaeyOeI0IaamOramaaBaaaleaacaWGobaabeaakmaabmaabaGa amiDaaGaayjkaiaawMcaaiaaykW7aiaawEa7caGLiWoadaahaaWcbe qaaiaaikdaaaaakiaawIcacaGLPaaacaaI9aWaaSaaaeaacaaIXaaa baGaamOtamaaCaaaleqabaGaaGOmaaaaaaGcdaaeqbqabSqaaiaadM gacaaISaGaamOAaiabgIGiolaadwfaaeqaniabggHiLdGcdaWcaaqa aiabec8aWnaaBaaaleaacaWGPbGaaGilaiaadQgaaeqaaOGaeyOeI0 IaeqiWda3aaSbaaSqaaiaadMgaaeqaaOGaeqiWda3aaSbaaSqaaiaa dQgaaeqaaaGcbaGaeqiWda3aaSbaaSqaaiaadMgaaeqaaOGaeqiWda 3aaSbaaSqaaiaadQgaaeqaaaaakmaadmaabaGaamysamaabmaabaGa amyEamaaBaaaleaacaWGPbaabeaakiabgsMiJkaadshaaiaawIcaca GLPaaacqGHsislceWGhbGbaebadaqhaaWcbaGaamyAaaqaaiaaiQca aaGcdaqadaqaaiaadshaaiaawIcacaGLPaaaaiaawUfacaGLDbaada WadaqaaiaadMeadaqadaqaaiaadMhadaWgaaWcbaGaamOAaaqabaGc cqGHKjYOcaWG0baacaGLOaGaayzkaaGaeyOeI0Iabm4rayaaraWaa0 baaSqaaiaadQgaaeaacaaIQaaaaOWaaeWaaeaacaWG0baacaGLOaGa ayzkaaaacaGLBbGaayzxaaGaey4kaSIaam4BamaabmaabaGaamOBam aaCaaaleqabaGaeyOeI0IaaGymaaaaaOGaayjkaiaawMcaaaaa@895C@

Based on this conjecture, we tested

V ˜ ( F ˜ * ( 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 ) G ˜ i * ( t ) ] [ I ( y j t ) G ˜ j * ( t ) ] . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOvayaaia WaaeWaaeaaceWGgbGbaGaadaahaaWcbeqaaiaaiQcaaaGcdaqadaqa aiaadshaaiaawIcacaGLPaaaaiaawIcacaGLPaaacaaI6aGaaGypam aalaaabaGaaGymaaqaaiaad6eadaahaaWcbeqaaiaaikdaaaaaaOWa aabuaeqaleaacaWGPbGaaGilaiaadQgacqGHiiIZcaWGZbaabeqdcq GHris5aOWaaSaaaeaacqaHapaCdaWgaaWcbaGaamyAaiaaiYcacaWG QbaabeaakiabgkHiTiabec8aWnaaBaaaleaacaWGPbaabeaakiabec 8aWnaaBaaaleaacaWGQbaabeaaaOqaaiabec8aWnaaBaaaleaacaWG PbGaaGilaiaadQgaaeqaaOGaeqiWda3aaSbaaSqaaiaadMgaaeqaaO GaeqiWda3aaSbaaSqaaiaadQgaaeqaaaaakmaadmaabaGaamysamaa bmaabaGaamyEamaaBaaaleaacaWGPbaabeaakiabgsMiJkaadshaai aawIcacaGLPaaacqGHsislceWGhbGbaGaadaqhaaWcbaGaamyAaaqa aiaaiQcaaaGcdaqadaqaaiaadshaaiaawIcacaGLPaaaaiaawUfaca GLDbaadaWadaqaaiaadMeadaqadaqaaiaadMhadaWgaaWcbaGaamOA aaqabaGccqGHKjYOcaWG0baacaGLOaGaayzkaaGaeyOeI0Iabm4ray aaiaWaa0baaSqaaiaadQgaaeaacaaIQaaaaOWaaeWaaeaacaWG0baa caGLOaGaayzkaaaacaGLBbGaayzxaaGaaGOlaaaa@7BD2@

as estimator for the design mean square error of the generalized difference estimator F ˜ * ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaia WaaWbaaSqabeaacaaIQaaaaOGaaGzaVpaabmaabaGaamiDaaGaayjk aiaawMcaaaaa@3AD0@ in the simulation study of the following section.

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