A comparison between nonparametric estimators for finite population distribution functions 3. Model-based properties

In this section we provide asymptotic expansions for the model bias and the model variance of the estimators introduced in the previous section. The expansions are based on the following assumptions:

H N , s ( x ) : = 1 n i s I ( x i x ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamisamaaBa aaleaacaWGobGaaiilaiaadohaaeqaaOWaaeWaaeaacaWG4baacaGL OaGaayzkaaGaaGOoaiaai2dadaWcaaqaaiaaigdaaeaacaWGUbaaam aaqafabaGaamysamaabmaabaGaamiEamaaBaaaleaacaWGPbaabeaa kiabgsMiJkaadIhaaiaawIcacaGLPaaaaSqaaiaadMgacqGHiiIZca WGZbaabeqdcqGHris5aaaa@4B02@

H N , s ¯ ( x ) := 1 N n i s I ( x i x ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamisamaaBa aaleaacaWGobGaaGilaiqadohagaqeaaqabaGcdaqadaqaaiaadIha aiaawIcacaGLPaaacaaI6aGaaGypamaalaaabaGaaGymaaqaaiaad6 eacqGHsislcaWGUbaaamaaqafabaGaamysamaabmaabaGaamiEamaa BaaaleaacaWGPbaabeaakiabgsMiJkaadIhaaiaawIcacaGLPaaaaS qaaiaadMgacqGHjiYZcaWGZbaabeqdcqGHris5aaaa@4CE2@

α  : = max { sup x [ a , b ] | H N , s ( x ) H s ( x ) | ,  sup x [ a , b ] | H N , s ¯ ( x ) H s ¯ ( x ) | } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySdeMaaG Ooaiaai2daciGGTbGaaiyyaiaacIhadaGadaqaamaawafabeWcbaGa amiEaiabgIGiopaadmaabaGaamyyaiaaiYcacaWGIbaacaGLBbGaay zxaaaabeGcbaGaci4CaiaacwhacaGGWbaaaiaaykW7daabdaqaaiaa ykW7caWGibWaaSbaaSqaaiaad6eacaaISaGaam4CaaqabaGcdaqada qaaiaadIhaaiaawIcacaGLPaaacqGHsislcaWGibWaaSbaaSqaaiaa dohaaeqaaOWaaeWaaeaacaWG4baacaGLOaGaayzkaaGaaGPaVdGaay 5bSlaawIa7aiaaiYcadaGfqbqabSqaaiaadIhacqGHiiIZdaWadaqa aiaadggacaaISaGaamOyaaGaay5waiaaw2faaaqabOqaaiGacohaca GG1bGaaiiCaaaadaabdaqaaiaaykW7caWGibWaaSbaaSqaaiaad6ea caaISaGabm4CayaaraaabeaakmaabmaabaGaamiEaaGaayjkaiaawM caaiabgkHiTiaadIeadaWgaaWcbaGabm4Cayaaraaabeaakmaabmaa baGaamiEaaGaayjkaiaawMcaaiaaykW7aiaawEa7caGLiWoacaaMc8 oacaGL7bGaayzFaaaaaa@7949@

| m( x )m( x 0 ) m ( x 0 )( x x 0 ) 1 2 m ′′ ( x 0 ) ( x x 0 ) 2 |C  | x x 0 | 2+δ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peee0hXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaabdaqaai aaykW7caWGTbWaaeWaaeaacaWG4baacaGLOaGaayzkaaGaeyOeI0Ia amyBamaabmaabaGaamiEamaaBaaaleaacaaIWaaabeaaaOGaayjkai aawMcaaiabgkHiTiqad2gagaqbamaabmaabaGaamiEamaaBaaaleaa caaIWaaabeaaaOGaayjkaiaawMcaamaabmaabaGaamiEaiabgkHiTi aadIhadaWgaaWcbaGaaGimaaqabaaakiaawIcacaGLPaaacqGHsisl daWcaaqaaiaaigdaaeaacaaIYaaaaiqad2gagaqbgaqbamaabmaaba GaamiEamaaBaaaleaacaaIWaaabeaaaOGaayjkaiaawMcaamaabmaa baGaamiEaiabgkHiTiaadIhadaWgaaWcbaGaaGimaaqabaaakiaawI cacaGLPaaadaahaaWcbeqaaiaaikdaaaGccaaMc8oacaGLhWUaayjc SdGaeyizImQaam4qamaaemaabaGaaGPaVlaadIhacqGHsislcaWG4b WaaSbaaSqaaiaaicdaaeqaaOGaaGPaVdGaay5bSlaawIa7amaaCaaa leqabaGaaGPaVlaaikdacqGHRaWkcqaH0oazaaaaaa@7083@

| G ( ε | x ) G ( ε 0 | x 0 ) G ( 1,0 ) ( ε 0 | x 0 ) ( ε ε 0 ) G ( 0,1 ) ( ε 0 | x 0 ) ( x x 0 ) 1 2 ( G ( 2,0 ) ( ε 0 | x 0 ) ( ε ε 0 ) 2 + 2 G ( 1,1 ) ( ε 0 | x 0 ) ( ε ε 0 ) ( x x 0 ) + G ( 0,2 ) ( ε 0 | x 0 ) ( x x 0 ) 2 ) | C ( | ε ε 0 | 2 + δ + | x x 0 | 2 + δ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaadaabda abaeqabaGaaGPaVlaadEeadaqadaqaamaaeiaabaGaeqyTduMaaGPa VdGaayjcSdGaamiEaaGaayjkaiaawMcaaiabgkHiTiaadEeadaqada qaamaaeiaabaGaeqyTdu2aaSbaaSqaaiaaicdaaeqaaaGccaGLiWoa caWG4bWaaSbaaSqaaiaaicdaaeqaaaGccaGLOaGaayzkaaGaeyOeI0 Iaam4ramaaCaaaleqabaWaaeWaaeaacaaIXaGaaGilaiaaicdaaiaa wIcacaGLPaaaaaGcdaqadaqaamaaeiaabaGaeqyTdu2aaSbaaSqaai aaicdaaeqaaaGccaGLiWoacaWG4bWaaSbaaSqaaiaaicdaaeqaaaGc caGLOaGaayzkaaWaaeWaaeaacqaH1oqzcqGHsislcqaH1oqzdaWgaa WcbaGaaGimaaqabaaakiaawIcacaGLPaaacqGHsislcaWGhbWaaWba aSqabeaadaqadaqaaiaaicdacaaISaGaaGymaaGaayjkaiaawMcaaa aakmaabmaabaWaaqGaaeaacqaH1oqzdaWgaaWcbaGaaGimaaqabaaa kiaawIa7aiaadIhadaWgaaWcbaGaaGimaaqabaaakiaawIcacaGLPa aadaqadaqaaiaadIhacqGHsislcaWG4bWaaSbaaSqaaiaaicdaaeqa aaGccaGLOaGaayzkaaaabaGaaGPaVlaaykW7cqGHsisldaWcaaqaai aaigdaaeaacaaIYaaaamaabmaabaGaam4ramaaCaaaleqabaWaaeWa aeaacaaIYaGaaGilaiaaicdaaiaawIcacaGLPaaaaaGcdaqadaqaam aaeiaabaGaeqyTdu2aaSbaaSqaaiaaicdaaeqaaaGccaGLiWoacaWG 4bWaaSbaaSqaaiaaicdaaeqaaaGccaGLOaGaayzkaaWaaeWaaeaacq aH1oqzcqGHsislcqaH1oqzdaWgaaWcbaGaaGimaaqabaaakiaawIca caGLPaaadaahaaWcbeqaaiaaikdaaaGccqGHRaWkcaaIYaGaam4ram aaCaaaleqabaWaaeWaaeaacaaIXaGaaGilaiaaigdaaiaawIcacaGL PaaaaaGcdaqadaqaamaaeiaabaGaeqyTdu2aaSbaaSqaaiaaicdaae qaaaGccaGLiWoacaWG4bWaaSbaaSqaaiaaicdaaeqaaaGccaGLOaGa ayzkaaWaaeWaaeaacqaH1oqzcqGHsislcqaH1oqzdaWgaaWcbaGaaG imaaqabaaakiaawIcacaGLPaaadaqadaqaaiaadIhacqGHsislcaWG 4bWaaSbaaSqaaiaaicdaaeqaaaGccaGLOaGaayzkaaGaey4kaSIaam 4ramaaCaaaleqabaWaaeWaaeaacaaIWaGaaGilaiaaikdaaiaawIca caGLPaaaaaGcdaqadaqaamaaeiaabaGaeqyTdu2aaSbaaSqaaiaaic daaeqaaaGccaGLiWoacaWG4bWaaSbaaSqaaiaaicdaaeqaaaGccaGL OaGaayzkaaWaaeWaaeaacaWG4bGaeyOeI0IaamiEamaaBaaaleaaca aIWaaabeaaaOGaayjkaiaawMcaamaaCaaaleqabaGaaGOmaaaaaOGa ayjkaiaawMcaaiaaykW7aaGaay5bSlaawIa7aaqaaiaaywW7caaMf8 UaeyizImQaam4qamaabmaabaWaaqWaaeaacaaMc8UaeqyTduMaeyOe I0IaeqyTdu2aaSbaaSqaaiaaicdaaeqaaaGccaGLhWUaayjcSdWaaW baaSqabeaacaaMc8UaaGOmaiabgUcaRiabes7aKbaakiabgUcaRmaa emaabaGaaGPaVlaadIhacqGHsislcaWG4bWaaSbaaSqaaiaaicdaae qaaaGccaGLhWUaayjcSdWaaWbaaSqabeaacaaMc8UaaGOmaiabgUca Riabes7aKbaaaOGaayjkaiaawMcaaaaaaa@E078@

G ( r , s ) ( ε | x ) := ∂  r + s G ( ε | x ) / ( ε r x s ) for r , s = 0,1,2. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4ramaaCa aaleqabaWaaeWaaeaacaWGYbGaaGilaiaadohaaiaawIcacaGLPaaa aaGcdaqadaqaamaaeiaabaGaeqyTduMaaGPaVdGaayjcSdGaamiEaa GaayjkaiaawMcaaiaaiQdacaaI9aWaaSGbaeaacqGHciITdaahaaWc beqaaiaadkhacqGHRaWkcaWGZbaaaOGaam4ramaabmaabaWaaqGaae aacqaH1oqzcaaMc8oacaGLiWoacaWG4baacaGLOaGaayzkaaaabaWa aeWaaeaacqGHciITcqaH1oqzdaahaaWcbeqaaiaadkhaaaGccqGHci ITcaWG4bWaaWbaaSqabeaacaWGZbaaaaGccaGLOaGaayzkaaaaaiaa ywW7caqGMbGaae4BaiaabkhacaaMf8UaamOCaiaaiYcacaWGZbGaaG ypaiaaicdacaaISaGaaGymaiaaiYcacaaIYaGaaGOlaaaa@66A9@

Assumption (C1) poses a restriction on how the sample and nonsample x i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBa aaleaacaWGPbaabeaakiabgkHiTaaa@380D@ values are generated. Together with assumption (C2) it makes sure that the estimation errors of the kernel density estimators for h s ( x ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiAamaaBa aaleaacaWGZbaabeaakmaabmaabaGaamiEaaGaayjkaiaawMcaaaaa @39A0@ and h s ¯ ( x ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiAamaaBa aaleaacaaMc8Uabm4CayaaraaabeaakmaabmaabaGaamiEaaGaayjk aiaawMcaaaaa@3B43@ go to zero uniformly for x [ a + λ , b λ ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabgI GiopaadmaabaGaamyyaiabgUcaRiabeU7aSjaaiYcacaWGIbGaeyOe I0Iaeq4UdWgacaGLBbGaayzxaaaaaa@412C@ and that they are uniformly bounded for x [ a , b ] . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabgI GiopaadmaabaGaamyyaiaaiYcacaWGIbaacaGLBbGaayzxaaGaaiOl aaaa@3CA7@ Replacing (C1) by more specific assumptions may allow for relaxing (C2) and for improving the uniform convergence rate for the estimation error of the kernel density estimators (see for example the results in Hansen 2008). Assumption (C3) is finally needed to make sure that the model mean square errors of the two estimators converge to zero. It can be relaxed at the cost of slowing down the convergence rates. In addition to assumptions (C1) to (C3) we shall also need the following assumption (C4) to make sure that the model mean square errors of the generalized difference estimators go to zero:

π i  : n * π ( x i ) j U π ( x j ) , i U , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiWda3aaS baaSqaaiaadMgaaeqaaOGaaGOoaiaai2dacaWGUbWaaWbaaSqabeaa caaIQaaaaOWaaSaaaeaacqaHapaCdaqadaqaaiaadIhadaWgaaWcba GaamyAaaqabaaakiaawIcacaGLPaaaaeaadaaeqbqaaiabec8aWnaa bmaabaGaamiEamaaBaaaleaacaWGQbaabeaaaOGaayjkaiaawMcaaa WcbaGaamOAaiabgIGiolaadwfaaeqaniabggHiLdaaaOGaaGilaiaa ywW7caaMf8UaaGzbVlaadMgacqGHiiIZcaWGvbGaaGilaaaa@5503@

Proposition 1. Under assumptions (C1) to (C3) it follows 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 s ¯ ( x ) d x + o ( λ 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqbaeaabiGaaa qaaiaadweadaqadaqaaiqadAeagaqcamaabmaabaGaamiDaaGaayjk aiaawMcaaiabgkHiTiaadAeadaWgaaWcbaGaamOtaaqabaGcdaqada qaaiaadshaaiaawIcacaGLPaaaaiaawIcacaGLPaaaaeaacaaI9aGa eq4UdW2aaWbaaSqabeaacaaIYaaaaOWaaSaaaeaacaWGobGaeyOeI0 IaamOBaaqaaiaad6eaaaWaaSaaaeaacqaH8oqBdaWgaaWcbaGaaGOm aaqabaaakeaacaaIYaGaeqiVd02aaSbaaSqaaiaaicdaaeqaaaaakm aapedabeWcbaGaamyyaaqaaiaadkgaa0Gaey4kIipakmaadeaabaGa am4ramaaCaaaleqabaWaaeWaaeaacaaIYaGaaGilaiaaicdaaiaawI cacaGLPaaaaaGcdaqadaqaamaaeiaabaGaamiDaiabgkHiTiaad2ga daqadaqaaiaadIhaaiaawIcacaGLPaaacaaMc8oacaGLiWoacaWG4b aacaGLOaGaayzkaaWaaeWaaeaaceWGTbGbauaadaqadaqaaiaadIha aiaawIcacaGLPaaaaiaawIcacaGLPaaadaahaaWcbeqaaiaaikdaaa GccqGHsislcaWGhbWaaWbaaSqabeaadaqadaqaaiaaigdacaaISaGa aGimaaGaayjkaiaawMcaaaaakmaabmaabaWaaqGaaeaacaWG0bGaey OeI0IaamyBamaabmaabaGaamiEaaGaayjkaiaawMcaaiaaykW7aiaa wIa7aiaadIhaaiaawIcacaGLPaaaceWGTbGbauGbauaadaqadaqaai aadIhaaiaawIcacaGLPaaaaiaawUfaaaqaaaqaamaadiaabaGaaGjb VlabgkHiTiaaikdacaWGhbWaaWbaaSqabeaadaqadaqaaiaaigdaca aISaGaaGymaaGaayjkaiaawMcaaaaakmaabmaabaWaaqGaaeaacaWG 0bGaeyOeI0IaamyBamaabmaabaGaamiEaaGaayjkaiaawMcaaiaayk W7aiaawIa7aiaadIhaaiaawIcacaGLPaaaceWGTbGbauaadaqadaqa aiaadIhaaiaawIcacaGLPaaacqGHRaWkcaWGhbWaaWbaaSqabeaada qadaqaaiaaicdacaaISaGaaGOmaaGaayjkaiaawMcaaaaakmaabmaa baWaaqGaaeaacaWG0bGaeyOeI0IaamyBamaabmaabaGaamiEaaGaay jkaiaawMcaaiaaykW7aiaawIa7aiaadIhaaiaawIcacaGLPaaaaiaa w2faaiaadIgadaWgaaWcbaGaaGPaVlqadohagaqeaaqabaGcdaqada qaaiaadIhaaiaawIcacaGLPaaacaWGKbGaamiEaiabgUcaRiaad+ga daqadaqaaiabeU7aSnaaCaaaleqabaGaaGOmaaaaaOGaayjkaiaawM caaaaaaaa@B253@

and

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 1 ) , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqbaeaabiGaaa qaaiaabAhacaqGHbGaaeOCamaabmaabaGabmOrayaajaWaaeWaaeaa caWG0baacaGLOaGaayzkaaGaeyOeI0IaamOramaaBaaaleaacaWGob aabeaakmaabmaabaGaamiDaaGaayjkaiaawMcaaaGaayjkaiaawMca aaqaaiaai2dadaWcaaqaaiaaigdaaeaacaWGUbaaamaabmaabaWaaS aaaeaacaWGobGaeyOeI0IaamOBaaqaaiaad6eaaaaacaGLOaGaayzk aaWaaWbaaSqabeaacaaIYaaaaOWaa8qmaeqaleaacaWGHbaabaGaam OyaaqdcqGHRiI8aOWaamWaaeaacaWGhbWaaeWaaeaadaabcaqaaiaa dshacqGHsislcaWGTbWaaeWaaeaacaWG4baacaGLOaGaayzkaaGaaG PaVdGaayjcSdGaamiEaaGaayjkaiaawMcaaiabgkHiTiaadEeadaah aaWcbeqaaiaaikdaaaGcdaqadaqaamaaeiaabaGaamiDaiabgkHiTi aad2gadaqadaqaaiaadIhaaiaawIcacaGLPaaacaaMc8oacaGLiWoa caWG4baacaGLOaGaayzkaaaacaGLBbGaayzxaaWaamWaaeaadaWcga qaaiaadIgadaWgaaWcbaGaaGPaVlqadohagaqeaaqabaGcdaqadaqa aiaadIhaaiaawIcacaGLPaaaaeaacaWGObWaaSbaaSqaaiaadohaae qaaOWaaeWaaeaacaWG4baacaGLOaGaayzkaaaaaaGaay5waiaaw2fa aiaadIgadaWgaaWcbaGaaGPaVlqadohagaqeaaqabaGcdaqadaqaai aadIhaaiaawIcacaGLPaaacaWGKbGaamiEaaqaaaqaaiaaysW7cqGH RaWkdaWcaaqaaiaaigdaaeaacaWGobGaeyOeI0IaamOBaaaadaqada qaamaalaaabaGaamOtaiabgkHiTiaad6gaaeaacaWGobaaaaGaayjk aiaawMcaamaaCaaaleqabaGaaGOmaaaakmaapedabeWcbaGaamyyaa qaaiaadkgaa0Gaey4kIipakmaadmaabaGaam4ramaabmaabaWaaqGa aeaacaWG0bGaeyOeI0IaamyBamaabmaabaGaamiEaaGaayjkaiaawM caaiaaykW7aiaawIa7aiaadIhaaiaawIcacaGLPaaacqGHsislcaWG hbWaaWbaaSqabeaacaaIYaaaaOWaaeWaaeaadaabcaqaaiaadshacq GHsislcaWGTbWaaeWaaeaacaWG4baacaGLOaGaayzkaaGaaGPaVdGa ayjcSdGaamiEaaGaayjkaiaawMcaaaGaay5waiaaw2faaiaadIgada WgaaWcbaGaaGPaVlqadohagaqeaaqabaGcdaqadaqaaiaadIhaaiaa wIcacaGLPaaacaWGKbGaamiEaiabgUcaRiaad+gadaqadaqaaiaad6 gadaahaaWcbeqaaiabgkHiTiaaigdaaaaakiaawIcacaGLPaaacaaI Saaaaaaa@B9B0@

where μ r := 1 1 K ( u ) u r d u MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiVd02aaS baaSqaaiaadkhaaeqaaOGaaGOoaiaai2dadaWdXaqabSqaaiabgkHi TiaaigdaaeaacqGHsislcaaIXaaaniabgUIiYdGccaWGlbWaaeWaae aacaWG1baacaGLOaGaayzkaaGaamyDamaaCaaaleqabaGaamOCaaaa kiaadsgacaWG1baaaa@464F@ for r = 0,1,2. MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCaiaai2 dacaaIWaGaaGilaiaaigdacaaISaGaaGOmaiaac6caaaa@3B0C@

Adding assumption (C4) it can be shown 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 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqbaeaabiGaaa qaaiaadweadaqadaqaaiqadAeagaacamaabmaabaGaamiDaaGaayjk aiaawMcaaiabgkHiTiaadAeadaWgaaWcbaGaamOtaaqabaGcdaqada qaaiaadshaaiaawIcacaGLPaaaaiaawIcacaGLPaaaaeaacaaI9aGa eq4UdW2aaWbaaSqabeaacaaIYaaaaOWaaSaaaeaacaWGobGaeyOeI0 IaamOBaaqaaiaad6eaaaWaaSaaaeaacqaH8oqBdaWgaaWcbaGaaGOm aaqabaaakeaacaaIYaGaeqiVd02aaSbaaSqaaiaaicdaaeqaaaaakm aapedabeWcbaGaamyyaaqaaiaadkgaa0Gaey4kIipakmaadeaabaGa am4ramaaCaaaleqabaWaaeWaaeaacaaIYaGaaGilaiaaicdaaiaawI cacaGLPaaaaaGcdaqadaqaamaaeiaabaGaamiDaiabgkHiTiaad2ga daqadaqaaiaadIhaaiaawIcacaGLPaaacaaMc8oacaGLiWoacaWG4b aacaGLOaGaayzkaaWaaeWaaeaaceWGTbGbauaadaqadaqaaiaadIha aiaawIcacaGLPaaaaiaawIcacaGLPaaadaahaaWcbeqaaiaaikdaaa GccqGHsislcaWGhbWaaWbaaSqabeaadaqadaqaaiaaigdacaaISaGa aGimaaGaayjkaiaawMcaaaaakmaabmaabaWaaqGaaeaacaWG0bGaey OeI0IaamyBamaabmaabaGaamiEaaGaayjkaiaawMcaaiaaykW7aiaa wIa7aiaadIhaaiaawIcacaGLPaaaceWGTbGbauGbauaadaqadaqaai aadIhaaiaawIcacaGLPaaaaiaawUfaaaqaaaqaaiaaysW7daWacaqa aiabgkHiTiaaikdacaWGhbWaaWbaaSqabeaadaqadaqaaiaaigdaca aISaGaaGymaaGaayjkaiaawMcaaaaakmaabmaabaWaaqGaaeaacaWG 0bGaeyOeI0IaamyBamaabmaabaGaamiEaaGaayjkaiaawMcaaiaayk W7aiaawIa7aiaadIhaaiaawIcacaGLPaaaceWGTbGbauaadaqadaqa aiaadIhaaiaawIcacaGLPaaacqGHRaWkcaWGhbWaaWbaaSqabeaada qadaqaaiaaicdacaaISaGaaGOmaaGaayjkaiaawMcaaaaakmaabmaa baWaaqGaaeaacaWG0bGaeyOeI0IaamyBamaabmaabaGaamiEaaGaay jkaiaawMcaaiaaykW7aiaawIa7aiaadIhaaiaawIcacaGLPaaaaiaa w2faaiaadIgadaqadaqaaiaadIhaaiaawIcacaGLPaaacaWGKbGaam iEaiabgUcaRiaad+gadaqadaqaaiabeU7aSnaaCaaaleqabaGaaGOm aaaaaOGaayjkaiaawMcaaiaaiYcaaaaaaa@B037@

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 daqaaiaadIhaaiaawIcacaGLPaaacaaISaaaaa@4FCC@

and it can be shown that

var ( F ˜ ( t ) F N ( t ) ) = var ( F ^ ( t ) F N ( t ) ) + o ( n 1 ) . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeODaiaabg gacaqGYbWaaeWaaeaaceWGgbGbaGaadaqadaqaaiaadshaaiaawIca caGLPaaacqGHsislcaWGgbWaaSbaaSqaaiaad6eaaeqaaOWaaeWaae aacaWG0baacaGLOaGaayzkaaaacaGLOaGaayzkaaGaaGypaiaabAha caqGHbGaaeOCamaabmaabaGabmOrayaajaWaaeWaaeaacaWG0baaca GLOaGaayzkaaGaeyOeI0IaamOramaaBaaaleaacaWGobaabeaakmaa bmaabaGaamiDaaGaayjkaiaawMcaaaGaayjkaiaawMcaaiabgUcaRi aad+gadaqadaqaaiaad6gadaahaaWcbeqaaiabgkHiTiaaigdaaaaa kiaawIcacaGLPaaacaaIUaaaaa@56A3@

Proposition 2. Under assumptions (C1) to (C3) and assuming that

σ 2 ( x ) := ε 2 d G ( ε | x ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaW baaSqabeaacaaIYaaaaOWaaeWaaeaacaWG4baacaGLOaGaayzkaaGa aGOoaiaai2dadaWdXaqaaiabew7aLnaaCaaaleqabaGaaGOmaaaaki aadsgacaWGhbWaaeWaaeaadaabcaqaaiabew7aLjaaykW7aiaawIa7 aiaadIhaaiaawIcacaGLPaaaaSqaaiabgkHiTiabg6HiLcqaaiabg6 HiLcqdcqGHRiI8aaaa@4D5A@

sup x [ a , b ] ε 4 d G ( ε | x ) < , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaybuaeqale aacaWG4bGaeyicI48aamWaaeaacaWGHbGaaGilaiaadkgaaiaawUfa caGLDbaaaeqakeaaciGGZbGaaiyDaiaacchaaaGaaGPaVpaapedaba GaeqyTdu2aaWbaaSqabeaacaaI0aaaaOGaamizaiaadEeadaqadaqa amaaeiaabaGaeqyTduMaaGPaVdGaayjcSdGaamiEaaGaayjkaiaawM caaiaaiYdacqGHEisPaSqaaiabgkHiTiabg6HiLcqaaiabg6HiLcqd cqGHRiI8aOGaaGilaaaa@5577@

it can be shown that

E ( F ^ * ( t ) F N ( t ) ) = λ 2 N n N μ 2 μ 0 a b G ( 0,2 ) ( t m ( x ) | x ) h s ¯ ( x ) d x + 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 1 ( x ) h s ¯ ( x ) d x + κ θ μ 0 2 a b G ( 2 , 0 ) ( t m ( x ) | x ) σ 2 ( x ) h s 1 ( x ) h s ¯ ( x ) d x ] + o ( λ 2 + ( n λ ) 1 ) , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqbaeaabmGaaa qaaiaadweadaqadaqaaiqadAeagaqcamaaCaaaleqabaGaaGOkaaaa kiaaygW7daqadaqaaiaadshaaiaawIcacaGLPaaacqGHsislcaWGgb WaaSbaaSqaaiaad6eaaeqaaOWaaeWaaeaacaWG0baacaGLOaGaayzk aaaacaGLOaGaayzkaaaabaGaaGypaiabeU7aSnaaCaaaleqabaGaaG OmaaaakmaalaaabaGaamOtaiabgkHiTiaad6gaaeaacaWGobaaamaa laaabaGaeqiVd02aaSbaaSqaaiaaikdaaeqaaaGcbaGaeqiVd02aaS baaSqaaiaaicdaaeqaaaaakmaapedabeWcbaGaamyyaaqaaiaadkga a0Gaey4kIipakiaadEeadaahaaWcbeqaamaabmaabaGaaGimaiaaiY cacaaIYaaacaGLOaGaayzkaaaaaOWaaeWaaeaadaabcaqaaiaadsha cqGHsislcaWGTbWaaeWaaeaacaWG4baacaGLOaGaayzkaaGaaGPaVd GaayjcSdGaamiEaaGaayjkaiaawMcaaiaadIgadaWgaaWcbaGaaGPa VlqadohagaqeaaqabaGcdaqadaqaaiaadIhaaiaawIcacaGLPaaaca WGKbGaamiEaaqaaaqaaiaaysW7cqGHRaWkdaWcaaqaaiaaigdaaeaa caWGUbGaeq4UdWgaamaalaaabaGaamOtaiabgkHiTiaad6gaaeaaca WGobaaamaadeaabaWaaSaaaeaacaWGlbWaaeWaaeaacaaIWaaacaGL OaGaayzkaaGaeyOeI0IaeqOUdSgabaGaeqiVd02aaSbaaSqaaiaaic daaeqaaaaakmaapedabeWcbaGaamyyaaqaaiaadkgaa0Gaey4kIipa kiaadEeadaahaaWcbeqaamaabmaabaGaaGymaiaaiYcacaaIWaaaca GLOaGaayzkaaaaaOWaaeWaaeaadaabcaqaaiaadshacqGHsislcaWG TbWaaeWaaeaacaWG4baacaGLOaGaayzkaaGaaGPaVdGaayjcSdGaam iEaaGaayjkaiaawMcaamaabmaabaGaamiDaiabgkHiTiaad2gadaqa daqaaiaadIhaaiaawIcacaGLPaaaaiaawIcacaGLPaaacaWGObWaa0 baaSqaaiaadohaaeaacqGHsislcaaIXaaaaOWaaeWaaeaacaWG4baa caGLOaGaayzkaaGaamiAamaaBaaaleaacaaMc8Uabm4Cayaaraaabe aakmaabmaabaGaamiEaaGaayjkaiaawMcaaiaadsgacaWG4baacaGL BbaaaeaaaeaacaaMe8+aamGaaeaacqGHRaWkdaWcaaqaaiabeQ7aRj abgkHiTiabeI7aXbqaaiabeY7aTnaaDaaaleaacaaIWaaabaGaaGOm aaaaaaGcdaWdXaqabSqaaiaadggaaeaacaWGIbaaniabgUIiYdGcca WGhbWaaWbaaSqabeaadaqadaqaaiaaikdacaGGSaGaaGimaaGaayjk aiaawMcaaaaakmaabmaabaWaaqGaaeaacaWG0bGaeyOeI0IaamyBam aabmaabaGaamiEaaGaayjkaiaawMcaaiaaykW7aiaawIa7aiaadIha aiaawIcacaGLPaaacqaHdpWCdaahaaWcbeqaaiaaikdaaaGcdaqada qaaiaadIhaaiaawIcacaGLPaaacaWGObWaa0baaSqaaiaadohaaeaa cqGHsislcaaIXaaaaOWaaeWaaeaacaWG4baacaGLOaGaayzkaaGaam iAamaaBaaaleaacaaMc8Uabm4CayaaraaabeaakmaabmaabaGaamiE aaGaayjkaiaawMcaaiaadsgacaWG4baacaGLDbaacqGHRaWkcaWGVb WaaeWaaeaacqaH7oaBdaahaaWcbeqaaiaaikdaaaGccqGHRaWkdaqa daqaaiaad6gacqaH7oaBaiaawIcacaGLPaaadaahaaWcbeqaaiabgk HiTiaaigdaaaaakiaawIcacaGLPaaacaaISaaaaaaa@E6FE@

where κ := 1 1 K 2 ( u ) d u MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOUdSMaaG Ooaiaai2dadaWdXaqaaiaadUeadaahaaWcbeqaaiaaikdaaaaabaGa eyOeI0IaaGymaaqaaiaaigdaa0Gaey4kIipakmaabmaabaGaamyDaa GaayjkaiaawMcaaiaadsgacaWG1baaaa@42E6@ and θ := 1 1 K ( v ) 1 1 K ( u + v ) K ( u ) d u d v , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiUdeNaaG Ooaiaai2dadaWdXaqaaiaadUeadaqadaqaaiaadAhaaiaawIcacaGL PaaadaWdXaqaaiaadUeacaaIOaGaamyDaiabgUcaRiaadAhacaaIPa Gaam4samaabmaabaGaamyDaaGaayjkaiaawMcaaiaadsgacaWG1bGa amizaiaadAhaaSqaaiabgkHiTiaaigdaaeaacaaIXaaaniabgUIiYd aaleaacqGHsislcaaIXaaabaGaaGymaaqdcqGHRiI8aOGaaiilaaaa @518C@ and it can be shown that

var ( F ^ * ( t ) F N ( t ) ) = var ( F ^ ( t ) F N ( t ) ) + o ( n 1 + λ 5 ) . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeODaiaabg gacaqGYbWaaeWaaeaaceWGgbGbaKaadaahaaWcbeqaaiaaiQcaaaGc caaMb8+aaeWaaeaacaWG0baacaGLOaGaayzkaaGaeyOeI0IaamOram aaBaaaleaacaWGobaabeaakmaabmaabaGaamiDaaGaayjkaiaawMca aaGaayjkaiaawMcaaiaai2dacaqG2bGaaeyyaiaabkhadaqadaqaai qadAeagaqcamaabmaabaGaamiDaaGaayjkaiaawMcaaiabgkHiTiaa dAeadaWgaaWcbaGaamOtaaqabaGcdaqadaqaaiaadshaaiaawIcaca GLPaaaaiaawIcacaGLPaaacqGHRaWkcaWGVbWaaeWaaeaacaWGUbWa aWbaaSqabeaacqGHsislcaaIXaaaaOGaey4kaSIaeq4UdW2aaWbaaS qabeaacaaI1aaaaaGccaGLOaGaayzkaaGaaGOlaaaa@5CA5@

Adding assumption (C4) it can also be shown that

E( F ˜ * ( t ) F N ( t ) ) = λ 2 Nn N μ 2 μ 0 a b G ( 0,2 ) ( tm( x )|x )h( x )dx + 1 nλ Nn N [ K( 0 )κ μ 0 a b G ( 1,0 ) ( tm( x )|x )( tm( x ) ) h s 1 ( x )h( x )dx + κθ μ 0 2 a b G ( 2,0 ) ( tm( x )|x ) σ 2 ( x ) h s 1 ( x )h( x )dx ] +o( λ 2 + ( nλ ) 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFjpeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peee0hXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaafaqaaeWaca aabaGaamyramaabmaabaGabmOrayaaiaWaaWbaaSqabeaacaaIQaaa aOWaaeWaaeaacaWG0baacaGLOaGaayzkaaGaeyOeI0IaamOramaaBa aaleaacaWGobaabeaakmaabmaabaGaamiDaaGaayjkaiaawMcaaaGa ayjkaiaawMcaaaqaaiaai2dacqaH7oaBdaahaaWcbeqaaiaaikdaaa GcdaWcaaqaaiaad6eacqGHsislcaWGUbaabaGaamOtaaaadaWcaaqa aiabeY7aTnaaBaaaleaacaaIYaaabeaaaOqaaiabeY7aTnaaBaaale aacaaIWaaabeaaaaGcdaWdXaqaaiaadEeadaahaaWcbeqaamaabmaa baGaaGimaiaaiYcacaaIYaaacaGLOaGaayzkaaaaaaqaaiaadggaae aacaWGIbaaniabgUIiYdGcdaqadaqaamaaeiaabaGaamiDaiabgkHi Tiaad2gadaqadaqaaiaadIhaaiaawIcacaGLPaaacaaMc8oacaGLiW oacaWG4baacaGLOaGaayzkaaGaamiAamaabmaabaGaamiEaaGaayjk aiaawMcaaiaadsgacaWG4baabaaabaGaaGjbVlabgUcaRmaalaaaba GaaGymaaqaaiaad6gacqaH7oaBaaWaaSaaaeaacaWGobGaeyOeI0Ia amOBaaqaaiaad6eaaaWaamqaaeaadaWcaaqaaiaadUeadaqadaqaai aaicdaaiaawIcacaGLPaaacqGHsislcqaH6oWAaeaacqaH8oqBdaWg aaWcbaGaaGimaaqabaaaaOWaa8qmaeaacaWGhbWaaWbaaSqabeaada qadaqaaiaaigdacaaISaGaaGimaaGaayjkaiaawMcaaaaaaeaacaWG HbaabaGaamOyaaqdcqGHRiI8aOWaaeWaaeaadaabcaqaaiaadshacq GHsislcaWGTbWaaeWaaeaacaWG4baacaGLOaGaayzkaaGaaGPaVdGa ayjcSdGaamiEaaGaayjkaiaawMcaamaabmaabaGaamiDaiabgkHiTi aad2gadaqadaqaaiaadIhaaiaawIcacaGLPaaaaiaawIcacaGLPaaa caWGObWaa0baaSqaaiaadohaaeaacqGHsislcaaIXaaaaOWaaeWaae aacaWG4baacaGLOaGaayzkaaGaamiAamaabmaabaGaamiEaaGaayjk aiaawMcaaiaadsgacaWG4baacaGLBbaaaeaaaqaabeqaaiaaysW7da WacaqaaiaaysW7caaMe8UaaGjbVlaaysW7caaMe8UaaGjbVlaaysW7 caaMe8UaaGjbVlaaysW7caaMe8UaaGjbVlaaysW7caaMe8UaaGjbVl aaysW7cqGHRaWkdaWcaaqaaiabeQ7aRjabgkHiTiabeI7aXbqaaiab eY7aTnaaDaaaleaacaaIWaaabaGaaGOmaaaaaaGcdaWdXaqaaiaadE eadaahaaWcbeqaamaabmaabaGaaGOmaiaaiYcacaaIWaaacaGLOaGa ayzkaaaaaaqaaiaadggaaeaacaWGIbaaniabgUIiYdGcdaqadaqaam aaeiaabaGaamiDaiabgkHiTiaad2gadaqadaqaaiaadIhaaiaawIca caGLPaaacaaMc8oacaGLiWoacaWG4baacaGLOaGaayzkaaGaeq4Wdm 3aaWbaaSqabeaacaaIYaaaaOWaaeWaaeaacaWG4baacaGLOaGaayzk aaGaamiAamaaDaaaleaacaWGZbaabaGaeyOeI0IaaGymaaaakmaabm aabaGaamiEaaGaayjkaiaawMcaaiaadIgadaqadaqaaiaadIhaaiaa wIcacaGLPaaacaWGKbGaamiEaaGaayzxaaaabaGaaGjbVlabgUcaRi aad+gadaqadaqaaiabeU7aSnaaCaaaleqabaGaaGOmaaaakiabgUca RmaabmaabaGaamOBaiabeU7aSbGaayjkaiaawMcaamaaCaaaleqaba GaeyOeI0IaaGymaaaaaOGaayjkaiaawMcaaaaaaaaa@F971@

and that

var ( F ˜ * ( t ) F N ( t ) ) = var ( F ^ ( t ) F N ( t ) ) + o ( n 1 + λ 5 ) . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeODaiaabg gacaqGYbWaaeWaaeaaceWGgbGbaGaadaahaaWcbeqaaiaaiQcaaaGc caaMb8+aaeWaaeaacaWG0baacaGLOaGaayzkaaGaeyOeI0IaamOram aaBaaaleaacaWGobaabeaakmaabmaabaGaamiDaaGaayjkaiaawMca aaGaayjkaiaawMcaaiaai2dacaqG2bGaaeyyaiaabkhadaqadaqaai qadAeagaqcamaabmaabaGaamiDaaGaayjkaiaawMcaaiabgkHiTiaa dAeadaWgaaWcbaGaamOtaaqabaGcdaqadaqaaiaadshaaiaawIcaca GLPaaaaiaawIcacaGLPaaacqGHRaWkcaWGVbWaaeWaaeaacaWGUbWa aWbaaSqabeaacqGHsislcaaIXaaaaOGaey4kaSIaeq4UdW2aaWbaaS qabeaacaaI1aaaaaGccaGLOaGaayzkaaGaaGOlaaaa@5CA4@

The proofs of the Propositions are given in the Appendix. Dorfman and Hall (1993) derived similar expansions for the Kuo estimator with local constant regression weights instead of local linear ones.

Note that in view of the asymptotic expansions it is possible to choose bandwidth sequences λ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4UdWgaaa@36B3@ in such a way as to make sure that the squares of the model biases are of smaller order of magnitude than the corresponding model variances. For the estimators based on the fitted values of Kuo this is achieved whenever λ = o ( n 1 / 4 ) , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4UdWMaaG ypaiaad+gadaqadaqaaiaad6gadaahaaWcbeqaaiabgkHiTmaalyaa baGaaGymaaqaaiaaisdaaaaaaaGccaGLOaGaayzkaaGaaiilaaaa@3E4D@ while for the estimators with the modified fitted values this requires that λ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4UdWgaaa@36B3@ goes to zero faster than O ( n 1 / 4 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4tamaabm aabaGaamOBamaaCaaaleqabaGaeyOeI0YaaSGbaeaacaaIXaaabaGa aGinaaaaaaaakiaawIcacaGLPaaaaaa@3B02@ and slower than O ( n 1 / 2 ) . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4tamaabm aabaGaamOBamaaCaaaleqabaGaeyOeI0YaaSGbaeaacaaIXaaabaGa aGOmaaaaaaaakiaawIcacaGLPaaacaGGUaaaaa@3BB2@ The convergence rates for the model biases of the latter estimators are optimized when λ = O ( n 1 / 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4UdWMaaG ypaiaad+eadaqadaqaaiaad6gadaahaaWcbeqaaiabgkHiTmaalyaa baGaaGymaaqaaiaaiodaaaaaaaGccaGLOaGaayzkaaaaaa@3D7C@ and in this case the resulting model biases are both of order O ( n 2 / 3 ) . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4tamaabm aabaGaamOBamaaCaaaleqabaGaeyOeI0YaaSGbaeaacaaIYaaabaGa aG4maaaaaaaakiaawIcacaGLPaaacaGGUaaaaa@3BB4@ The model biases for the estimators based on the fitted values of Kuo can be made to converge much faster, depending on the sequences H N , s ( x ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamisamaaBa aaleaacaWGobGaaGilaiaadohaaeqaaOWaaeWaaeaacaWG4baacaGL OaGaayzkaaaaaa@3B09@ and H N , s ¯ ( x ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamisamaaBa aaleaacaWGobGaaGilaiaaykW7ceWGZbGbaebaaeqaaOWaaeWaaeaa caWG4baacaGLOaGaayzkaaaaaa@3CAC@ and on the bandwidth sequence  λ . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4UdWMaai Olaaaa@3765@

Given the above considerations concerning the model biases and given the fact that the leading terms in the model variances are the same for both types of fitted values, it would be of interest to know the second order terms in the model variances in order to establish which estimator is more efficient from the model-based perspective. The proofs in the Appendix suggest however that the second order terms depend on more specific assumptions than (C1) to (C3) and that, in particular for the estimators based on the modified fitted values, they are difficult to determine.

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