A comparison between nonparametric estimators for finite population distribution functions 2. Definition of the estimators

Let ( y i , x i ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaaca WG5bWaaSbaaSqaaiaadMgaaeqaaOGaaGilaiaadIhadaWgaaWcbaGa amyAaaqabaaakiaawIcacaGLPaaaaaa@3B81@ denote the values taken on by a study variable Y MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamywaaaa@35DD@ and an auxiliary variable X MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiwaaaa@35DC@ on unit i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAaaaa@35ED@ of a finite population U := { 1,2, , N } . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyvaiaaiQ dacaaI9aWaaiWaaeaacaaIXaGaaGilaiaaikdacaaISaGaeSOjGSKa aGilaiaad6eaaiaawUhacaGL9baacaGGUaaaaa@3FD5@ Suppose that

y i = m ( x i ) + ε i , i U , ( 2.1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEamaaBa aaleaacaWGPbaabeaakiaai2dacaWGTbWaaeWaaeaacaWG4bWaaSba aSqaaiaadMgaaeqaaaGccaGLOaGaayzkaaGaey4kaSIaeqyTdu2aaS baaSqaaiaadMgaaeqaaOGaaGilaiaaywW7caaMf8UaamyAaiabgIGi olaadwfacaaISaGaaGzbVlaaywW7caaMf8UaaGzbVlaaywW7caGGOa GaaGOmaiaac6cacaaIXaGaaiykaaaa@534C@

where m ( x ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBamaabm aabaGaamiEaaGaayjkaiaawMcaaaaa@3877@ is a smooth function and where the ε i s MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyTdu2aaS baaSqaaiaadMgaaeqaaGqaaOGaa8xgGiaabohaaaa@3983@ are independent zero mean random variables whose distribution functions P ( ε i ε ) = G ( ε | x i ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiuamaabm aabaGaeqyTdu2aaSbaaSqaaiaadMgaaeqaaOGaeyizImQaeqyTduga caGLOaGaayzkaaGaaGypaiaadEeadaqadaqaamaaeiaabaGaeqyTdu MaaGPaVdGaayjcSdGaamiEamaaBaaaleaacaWGPbaabeaaaOGaayjk aiaawMcaaaaa@4789@ depend smoothly on x i . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBa aaleaacaWGPbaabeaakiaac6caaaa@37D2@ Let s U MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4Caiabgk Oimlaadwfaaaa@38CD@ be a sample chosen from the population U MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyvaaaa@35D9@ according to some sample design. As usual in the context of complete auxiliary information we assume that the x i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBa aaleaacaWGPbaabeaakiaaykW7cqGHsislaaa@3998@ values are known for all population units, while the y i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEamaaBa aaleaacaWGPbaabeaakiaaykW7cqGHsislaaa@3999@ values are observed only for the population units which belong to the sample s . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4Caiaac6 caaaa@36A9@

To estimate the unknown population distribution function

F N ( t ) : = 1 N i U I ( y i t ) , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOramaaBa aaleaacaWGobaabeaakmaabmaabaGaamiDaaGaayjkaiaawMcaaiaa iQdacaaI9aWaaSaaaeaacaaIXaaabaGaamOtaaaadaaeqbqaaiaadM eadaqadaqaaiaadMhadaWgaaWcbaGaamyAaaqabaGccqGHKjYOcaWG 0baacaGLOaGaayzkaaaaleaacaWGPbGaeyicI4Saamyvaaqab0Gaey yeIuoakiaaiYcaaaa@49D3@

Kuo (1988) proposes the estimator given by

F ^ ( t ) : = 1 N ( j s I ( y j t ) + i s j s w i , j I ( y j t ) ) , ( 2.2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaja WaaeWaaeaacaWG0baacaGLOaGaayzkaaGaaGOoaiaai2dadaWcaaqa aiaaigdaaeaacaWGobaaamaabmaabaWaaabuaeaacaWGjbWaaeWaae aacaWG5bWaaSbaaSqaaiaadQgaaeqaaOGaeyizImQaamiDaaGaayjk aiaawMcaaaWcbaGaamOAaiabgIGiolaadohaaeqaniabggHiLdGccq GHRaWkdaaeqbqabSqaaiaadMgacqGHjiYZcaWGZbaabeqdcqGHris5 aOWaaabuaeaacaWG3bWaaSbaaSqaaiaadMgacaaISaGaamOAaaqaba GccaWGjbWaaeWaaeaacaWG5bWaaSbaaSqaaiaadQgaaeqaaOGaeyiz ImQaamiDaaGaayjkaiaawMcaaaWcbaGaamOAaiabgIGiolaadohaae qaniabggHiLdaakiaawIcacaGLPaaacaaISaGaaGzbVlaaywW7caaM f8UaaGzbVlaaywW7caGGOaGaaGOmaiaac6cacaaIYaGaaiykaaaa@6CCD@

where in place of w i , j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4DamaaBa aaleaacaWGPbGaaGilaiaadQgaaeqaaaaa@38BA@ she suggests to use either the local constant regression weights

w i,j := K( x i x j λ ) ks K( x i x k λ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peee0hXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWG3bWaaS baaSqaaiaadMgacaaISaGaamOAaaqabaGccaaMe8UaaGOoaiaai2da daWcaaqaaiaadUeadaqadaqaamaalaaabaGaamiEamaaBaaaleaaca WGPbaabeaakiabgkHiTiaadIhadaWgaaWcbaGaamOAaaqabaaakeaa cqaH7oaBaaaacaGLOaGaayzkaaaabaWaaabuaeaacaWGlbWaaeWaae aadaWcaaqaaiaadIhadaWgaaWcbaGaamyAaaqabaGccqGHsislcaWG 4bWaaSbaaSqaaiaadUgaaeqaaaGcbaGaeq4UdWgaaaGaayjkaiaawM caaaWcbaGaam4AaiabgIGiolaadohaaeqaniabggHiLdaaaaaa@5707@

with some (integrable) kernel function in place of K ( u ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4samaabm aabaGaamyDaaGaayjkaiaawMcaaaaa@3852@ and λ > 0 , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4UdWMaaG OpaiaaicdacaGGSaaaaa@38E5@ or the nearest k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4Aaaaa@35EF@ neighbor weights

w i , j  : = { 1 / k , if   x j  is one of the  k  nearest neighbors to  x i 0 , otherwise . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4DamaaBa aaleaacaWGPbGaaGilaiaadQgaaeqaaOGaaGOoaiaai2dadaqabaqa auaabaqaciaaaeaadaWcgaqaaiaaigdaaeaacaWGRbaaaiaacYcaae aacaqGPbGaaeOzaiaabccacaqGGaGaamiEamaaBaaaleaacaWGQbaa beaakiaabccacaqGPbGaae4CaiaabccacaqGVbGaaeOBaiaabwgaca qGGaGaae4BaiaabAgacaqGGaGaaeiDaiaabIgacaqGLbGaaeiiaiaa dUgacaqGGaGaaeOBaiaabwgacaqGHbGaaeOCaiaabwgacaqGZbGaae iDaiaabccacaqGUbGaaeyzaiaabMgacaqGNbGaaeiAaiaabkgacaqG VbGaaeOCaiaabohacaqGGaGaaeiDaiaab+gacaqGGaGaamiEamaaBa aaleaacaWGPbaabeaaaOqaaiaaicdacaGGSaaabaGaae4Baiaabsha caqGObGaaeyzaiaabkhacaqG3bGaaeyAaiaabohacaqGLbGaaeOlaa aaaiaawUhaaaaa@709E@

Note that in the definition F ^ ( t ) , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaja WaaeWaaeaacaWG0baacaGLOaGaayzkaaGaaiilaaaa@390C@

G ^ i ( t ) := j s w i , j I ( y j t ) ( 2.3 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabm4rayaaja WaaSbaaSqaaiaadMgaaeqaaOWaaeWaaeaacaWG0baacaGLOaGaayzk aaGaaGOoaiaai2dadaaeqbqaaiaadEhadaWgaaWcbaGaamyAaiaaiY cacaWGQbaabeaakiaadMeadaqadaqaaiaadMhadaWgaaWcbaGaamOA aaqabaGccqGHKjYOcaWG0baacaGLOaGaayzkaaaaleaacaWGQbGaey icI4Saam4Caaqab0GaeyyeIuoakiaaywW7caaMf8UaaGzbVlaaywW7 caaMf8UaaiikaiaaikdacaGGUaGaaG4maiaacMcaaaa@56DA@

is used as the fitted value in place of the unobserved indicator function I ( y i t ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysamaabm aabaGaamyEamaaBaaaleaacaWGPbaabeaakiabgsMiJkaadshaaiaa wIcacaGLPaaaaaa@3C26@ for i s . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAaiabgM GiplaadohacaGGUaaaaa@391D@

Following an idea put forward in the textbook of Chambers and Clark (2012), we shall analyze an estimator for F N ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOramaaBa aaleaacaWGobaabeaakmaabmaabaGaamiDaaGaayjkaiaawMcaaaaa @3955@ based on alternative fitted values which incorporate a nonparametric estimate for the mean regression function m ( x ) . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBamaabm aabaGaamiEaaGaayjkaiaawMcaaiaac6caaaa@3929@ The fitted values in question are given by

G ^ i * ( t ) := j s w i , j I ( y j m ^ j t m ^ i ) ( 2.4 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabm4rayaaja Waa0baaSqaaiaadMgaaeaacaaIQaaaaOWaaeWaaeaacaWG0baacaGL OaGaayzkaaGaaGOoaiaai2dadaaeqbqaaiaadEhadaWgaaWcbaGaam yAaiaaiYcacaWGQbaabeaakiaadMeadaqadaqaaiaadMhadaWgaaWc baGaamOAaaqabaGccqGHsislceWGTbGbaKaadaWgaaWcbaGaamOAaa qabaGccqGHKjYOcaWG0bGaeyOeI0IabmyBayaajaWaaSbaaSqaaiaa dMgaaeqaaaGccaGLOaGaayzkaaaaleaacaWGQbGaeyicI4Saam4Caa qab0GaeyyeIuoakiaaywW7caaMf8UaaGzbVlaaywW7caaMf8Uaaiik aiaaikdacaGGUaGaaGinaiaacMcaaaa@5DB7@

where

m ^ i := k s w i , j y j MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmyBayaaja WaaSbaaSqaaiaadMgaaeqaaOGaaGOoaiaai2dadaaeqbqaaiaadEha daWgaaWcbaGaamyAaiaaiYcacaWGQbaabeaakiaadMhadaWgaaWcba GaamOAaaqabaaabaGaam4AaiabgIGiolaadohaaeqaniabggHiLdaa aa@4411@

is a nonparametric estimator for m ( x ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBamaabm aabaGaamiEaaGaayjkaiaawMcaaaaa@3877@ at x = x i , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiaai2 dacaWG4bWaaSbaaSqaaiaadMgaaeqaaOGaaiilaaaa@3994@ and the resulting estimator for F N ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOramaaBa aaleaacaWGobaabeaakmaabmaabaGaamiDaaGaayjkaiaawMcaaaaa @3955@ is given by

F ^ * ( t ) := 1 N ( j s I ( y j t ) + i s j s w i , j I ( y j m ^ j t m ^ i ) ) . ( 2.5 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaja WaaWbaaSqabeaacaaIQaaaaOWaaeWaaeaacaWG0baacaGLOaGaayzk aaGaaGOoaiaai2dadaWcaaqaaiaaigdaaeaacaWGobaaamaabmaaba WaaabuaeaacaWGjbWaaeWaaeaacaWG5bWaaSbaaSqaaiaadQgaaeqa aOGaeyizImQaamiDaaGaayjkaiaawMcaaaWcbaGaamOAaiabgIGiol aadohaaeqaniabggHiLdGccqGHRaWkdaaeqbqabSqaaiaadMgacqGH jiYZcaWGZbaabeqdcqGHris5aOWaaabuaeaacaWG3bWaaSbaaSqaai aadMgacaaISaGaamOAaaqabaGccaWGjbWaaeWaaeaacaWG5bWaaSba aSqaaiaadQgaaeqaaOGaeyOeI0IabmyBayaajaWaaSbaaSqaaiaadQ gaaeqaaOGaeyizImQaamiDaiabgkHiTiqad2gagaqcamaaBaaaleaa caWGPbaabeaaaOGaayjkaiaawMcaaaWcbaGaamOAaiabgIGiolaado haaeqaniabggHiLdaakiaawIcacaGLPaaacaaIUaGaaGzbVlaaywW7 caaMf8UaaGzbVlaaywW7caGGOaGaaGOmaiaac6cacaaI1aGaaiykaa aa@73E4@

The fitted values in (2.3) and (2.4), or appropriately modified versions of them which include sample inclusion probabilities in the regression weights w i , j , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4DamaaBa aaleaacaWGPbGaaGilaiaadQgaaeqaaOGaaiilaaaa@3974@ can obviously be computed also for i s , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAaiabgI GiolaadohacaGGSaaaaa@3919@ and they can be employed for example in generalized difference estimators (Särndal et al. 1992, page 221) or in model calibrated estimators (see for example Wu and Sitter 2001; Chen and Wu 2002; Wu 2003; Montanari and Ranalli 2005; Rueda, Martínez, Martínez and Arcos 2007; Rueda, Sànchez-Borrego, Arcos and Martínez 2010). In addition to the model-based estimators in (2.2) and (2.5), we shall thus consider also the generalized difference estimators given by

F ˜ ( t ) := 1 N ( i U j s w ˜ i , j I ( y j t ) ) + i s π i 1 ( I ( y i t ) j s w ˜ i , j I ( y j t ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaia WaaeWaaeaacaWG0baacaGLOaGaayzkaaGaaGOoaiaai2dadaWcaaqa aiaaigdaaeaacaWGobaaamaabmaabaWaaabuaeqaleaacaWGPbGaey icI4Saamyvaaqab0GaeyyeIuoakmaaqafabaGabm4DayaaiaWaaSba aSqaaiaadMgacaaISaGaamOAaaqabaGccaWGjbWaaeWaaeaacaWG5b WaaSbaaSqaaiaadQgaaeqaaOGaeyizImQaamiDaaGaayjkaiaawMca aaWcbaGaamOAaiabgIGiolaadohaaeqaniabggHiLdaakiaawIcaca GLPaaacqGHRaWkdaaeqbqaaiabec8aWnaaDaaaleaacaWGPbaabaGa eyOeI0IaaGymaaaaaeaacaWGPbGaeyicI4Saam4Caaqab0GaeyyeIu oakmaabmaabaGaamysamaabmaabaGaamyEamaaBaaaleaacaWGPbaa beaakiabgsMiJkaadshaaiaawIcacaGLPaaacqGHsisldaaeqbqaai qadEhagaacamaaBaaaleaacaWGPbGaaGilaiaadQgaaeqaaOGaamys amaabmaabaGaamyEamaaBaaaleaacaWGQbaabeaakiabgsMiJkaads haaiaawIcacaGLPaaaaSqaaiaadQgacqGHiiIZcaWGZbaabeqdcqGH ris5aaGccaGLOaGaayzkaaaaaa@7839@

and by

F ˜ * ( t ) := 1 N ( i U j s w ˜ i , j I ( y j m ˜ j t m ˜ i ) ) + i s π i 1 ( I ( y i t ) j s w ˜ i , j I ( y j m ˜ j t m ˜ i ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaia WaaWbaaSqabeaacaaIQaaaaOWaaeWaaeaacaWG0baacaGLOaGaayzk aaGaaGOoaiaai2dadaWcaaqaaiaaigdaaeaacaWGobaaamaabmaaba WaaabuaeqaleaacaWGPbGaeyicI4Saamyvaaqab0GaeyyeIuoakmaa qafabaGabm4DayaaiaWaaSbaaSqaaiaadMgacaaISaGaamOAaaqaba GccaWGjbWaaeWaaeaacaWG5bWaaSbaaSqaaiaadQgaaeqaaOGaeyOe I0IabmyBayaaiaWaaSbaaSqaaiaadQgaaeqaaOGaeyizImQaamiDai abgkHiTiqad2gagaacamaaBaaaleaacaWGPbaabeaaaOGaayjkaiaa wMcaaaWcbaGaamOAaiabgIGiolaadohaaeqaniabggHiLdaakiaawI cacaGLPaaacqGHRaWkdaaeqbqaaiabec8aWnaaDaaaleaacaWGPbaa baGaeyOeI0IaaGymaaaakmaabmaabaGaamysamaabmaabaGaamyEam aaBaaaleaacaWGPbaabeaakiabgsMiJkaadshaaiaawIcacaGLPaaa cqGHsisldaaeqbqaaiqadEhagaacamaaBaaaleaacaWGPbGaaGilai aadQgaaeqaaOGaamysamaabmaabaGaamyEamaaBaaaleaacaWGQbaa beaakiabgkHiTiqad2gagaacamaaBaaaleaacaWGQbaabeaakiabgs MiJkaadshacqGHsislceWGTbGbaGaadaWgaaWcbaGaamyAaaqabaaa kiaawIcacaGLPaaaaSqaaiaadQgacqGHiiIZcaWGZbaabeqdcqGHri s5aaGccaGLOaGaayzkaaaaleaacaWGPbGaeyicI4Saam4Caaqab0Ga eyyeIuoaaaa@8579@

where π i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiWda3aaS baaSqaaiaadMgaaeqaaaaa@37D6@ denotes the first order sample inclusion probabilities, w ˜ i , j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabm4Dayaaia WaaSbaaSqaaiaadMgacaaISaGaamOAaaqabaaaaa@38C9@ denotes design weighted regression weights whose definition is given below, and m ˜ i := k s w ˜ i , k y k . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmyBayaaia WaaSbaaSqaaiaadMgaaeqaaOGaaGOoaiaai2dadaaeqaqabSqaaiaa dUgacqGHiiIZcaWGZbaabeqdcqGHris5aOGabm4DayaaiaWaaSbaaS qaaiaadMgacaaISaGaam4AaaqabaGccaWG5bWaaSbaaSqaaiaadUga aeqaaOGaaiOlaaaa@44B4@ Note that 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@ are based on design weighted counterparts of the fitted values G ^ i ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabm4rayaaja WaaSbaaSqaaiaadMgaaeqaaOWaaeWaaeaacaWG0baacaGLOaGaayzk aaaaaa@3981@ and G ^ i * ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabm4rayaaja Waa0baaSqaaiaadMgaaeaacaaIQaaaaOWaaeWaaeaacaWG0baacaGL OaGaayzkaaaaaa@3A36@ which are given by

G ˜ i ( t ) := j s w ˜ i , j I ( y j t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabm4rayaaia WaaSbaaSqaaiaadMgaaeqaaOWaaeWaaeaacaWG0baacaGLOaGaayzk aaGaaGOoaiaai2dadaaeqbqaaiqadEhagaacamaaBaaaleaacaWGPb GaaGilaiaadQgaaeqaaOGaamysamaabmaabaGaamyEamaaBaaaleaa caWGQbaabeaakiabgsMiJkaadshaaiaawIcacaGLPaaaaSqaaiaadQ gacqGHiiIZcaWGZbaabeqdcqGHris5aaaa@4B94@

and

G ˜ i * ( t ) := j s w ˜ i , j I ( y j m ˜ j t m ˜ i ) , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabm4rayaaia Waa0baaSqaaiaadMgaaeaacaaIQaaaaOWaaeWaaeaacaWG0baacaGL OaGaayzkaaGaaGOoaiaai2dadaaeqbqaaiqadEhagaacamaaBaaale aacaWGPbGaaGilaiaadQgaaeqaaOGaamysamaabmaabaGaamyEamaa BaaaleaacaWGQbaabeaakiabgkHiTiqad2gagaacamaaBaaaleaaca WGQbaabeaakiabgsMiJkaadshacqGHsislceWGTbGbaGaadaWgaaWc baGaamyAaaqabaaakiaawIcacaGLPaaaaSqaaiaadQgacqGHiiIZca WGZbaabeqdcqGHris5aOGaaGilaaaa@532E@

respectively.

As for the regression weights w i , j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4DamaaBa aaleaacaWGPbGaaGilaiaadQgaaeqaaaaa@38BA@ and w ˜ i , j , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabm4Dayaaia WaaSbaaSqaaiaadMgacaaISaGaamOAaaqabaGccaGGSaaaaa@3983@ in the present work we consider local linear regression weights in their place. In what follows w i , j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4DamaaBa aaleaacaWGPbGaaGilaiaadQgaaeqaaaaa@38BA@ and w ˜ i , j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabm4Dayaaia WaaSbaaSqaaiaadMgacaaISaGaamOAaaqabaaaaa@38C9@ are thus defined 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=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4DamaaBa aaleaacaWGPbGaaGilaiaadQgaaeqaaOGaaGOoaiaai2dadaWcaaqa aiaaigdaaeaacaWGUbGaeq4UdWgaaiaadUeadaqadaqaamaalaaaba GaamiEamaaBaaaleaacaWGPbaabeaakiabgkHiTiaadIhadaWgaaWc baGaamOAaaqabaaakeaacqaH7oaBaaaacaGLOaGaayzkaaWaaSaaae aacaWGnbWaaSbaaSqaaiaaikdacaaISaGaam4CaaqabaGcdaqadaqa aiaadIhadaWgaaWcbaGaamyAaaqabaaakiaawIcacaGLPaaacqGHsi sldaqadaqaamaalaaabaGaamiEamaaBaaaleaacaWGPbaabeaakiab gkHiTiaadIhadaWgaaWcbaGaamOAaaqabaaakeaacqaH7oaBaaaaca GLOaGaayzkaaGaamytamaaBaaaleaacaaIXaGaaGilaiaadohaaeqa aOWaaeWaaeaacaWG4bWaaSbaaSqaaiaadMgaaeqaaaGccaGLOaGaay zkaaaabaGaamytamaaBaaaleaacaaIYaGaaGilaiaadohaaeqaaOWa aeWaaeaacaWG4bWaaSbaaSqaaiaadMgaaeqaaaGccaGLOaGaayzkaa GaamytamaaBaaaleaacaaIWaGaaGilaiaadohaaeqaaOWaaeWaaeaa caWG4bWaaSbaaSqaaiaadMgaaeqaaaGccaGLOaGaayzkaaGaeyOeI0 IaamytamaaDaaaleaacaaIXaGaaGilaiaadohaaeaacaaIYaaaaOWa aeWaaeaacaWG4bWaaSbaaSqaaiaadMgaaeqaaaGccaGLOaGaayzkaa aaaaaa@75B9@

and

w ˜ i , j := 1 π j 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=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabm4Dayaaia WaaSbaaSqaaiaadMgacaaISaGaamOAaaqabaGccaaI6aGaaGypamaa laaabaGaaGymaaqaaiabec8aWnaaBaaaleaacaWGQbaabeaakiaad6 gacqaH7oaBaaGaam4samaabmaabaWaaSaaaeaacaWG4bWaaSbaaSqa aiaadMgaaeqaaOGaeyOeI0IaamiEamaaBaaaleaacaWGQbaabeaaaO qaaiabeU7aSbaaaiaawIcacaGLPaaadaWcaaqaaiqad2eagaacamaa BaaaleaacaaIYaGaaGilaiaadohaaeqaaOWaaeWaaeaacaWG4bWaaS baaSqaaiaadMgaaeqaaaGccaGLOaGaayzkaaGaeyOeI0YaaeWaaeaa daWcaaqaaiaadIhadaWgaaWcbaGaamyAaaqabaGccqGHsislcaWG4b WaaSbaaSqaaiaadQgaaeqaaaGcbaGaeq4UdWgaaaGaayjkaiaawMca aiqad2eagaacamaaBaaaleaacaaIXaGaaGilaiaadohaaeqaaOWaae WaaeaacaWG4bWaaSbaaSqaaiaadMgaaeqaaaGccaGLOaGaayzkaaaa baGabmytayaaiaWaaSbaaSqaaiaaikdacaaISaGaam4CaaqabaGcda qadaqaaiaadIhadaWgaaWcbaGaamyAaaqabaaakiaawIcacaGLPaaa ceWGnbGbaGaadaWgaaWcbaGaaGimaiaaiYcacaWGZbaabeaakmaabm aabaGaamiEamaaBaaaleaacaWGPbaabeaaaOGaayjkaiaawMcaaiab gkHiTiqad2eagaacamaaDaaaleaacaaIXaGaaGilaiaadohaaeaaca aIYaaaaOWaaeWaaeaacaWG4bWaaSbaaSqaaiaadMgaaeqaaaGccaGL OaGaayzkaaaaaiaaiYcaaaa@79AB@

where n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaaaa@35F2@ is the number of units in the sample s , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4CaiaacY caaaa@36A7@

M r , s ( x ) := k s 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=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamytamaaBa aaleaacaWGYbGaaGilaiaadohaaeqaaOWaaeWaaeaacaWG4baacaGL OaGaayzkaaGaaGOoaiaai2dadaaeqbqabSqaaiaadUgacqGHiiIZca WGZbaabeqdcqGHris5aOWaaSaaaeaacaaIXaaabaGaamOBaiabeU7a SbaacaWGlbWaaeWaaeaadaWcaaqaaiaadIhacqGHsislcaWG4bWaaS baaSqaaiaadUgaaeqaaaGcbaGaeq4UdWgaaaGaayjkaiaawMcaamaa bmaabaWaaSaaaeaacaWG4bGaeyOeI0IaamiEamaaBaaaleaacaWGRb aabeaaaOqaaiabeU7aSbaaaiaawIcacaGLPaaadaahaaWcbeqaaiaa dkhaaaGccaaISaGaaGzbVlaaywW7caaMf8UaamOCaiaai2dacaaIWa GaaGilaiaaigdacaaISaGaaGOmaiaaiYcaaaa@61EB@

and

M ˜ r , s ( x ) := k s 1 π k 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=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmytayaaia WaaSbaaSqaaiaadkhacaaISaGaam4CaaqabaGcdaqadaqaaiaadIha aiaawIcacaGLPaaacaaI6aGaaGypamaaqafabaWaaSaaaeaacaaIXa aabaGaeqiWda3aaSbaaSqaaiaadUgaaeqaaOGaamOBaiabeU7aSbaa caWGlbaaleaacaWGRbGaeyicI4Saam4Caaqab0GaeyyeIuoakmaabm aabaWaaSaaaeaacaWG4bGaeyOeI0IaamiEamaaBaaaleaacaWGRbaa beaaaOqaaiabeU7aSbaaaiaawIcacaGLPaaadaqadaqaamaalaaaba GaamiEaiabgkHiTiaadIhadaWgaaWcbaGaam4AaaqabaaakeaacqaH 7oaBaaaacaGLOaGaayzkaaWaaWbaaSqabeaacaWGYbaaaOGaaGilai aaywW7caaMf8UaaGzbVlaadkhacaaI9aGaaGimaiaaiYcacaaIXaGa aGilaiaaikdacaaIUaaaaa@64DE@

It is worth noting that the nonparametric estimators of this section are not well-defined if the regression weights w i , j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4DamaaBa aaleaacaWGPbGaaGilaiaadQgaaeqaaaaa@38BA@ and w ˜ i , j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabm4Dayaaia WaaSbaaSqaaiaadMgacaaISaGaamOAaaqabaaaaa@38C9@ included in their definitions are not well-defined. This problem occurs for example when the support of the kernel function K ( u ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4samaabm aabaGaamyDaaGaayjkaiaawMcaaaaa@3852@ is given by the interval [ 1,1 ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaamWaaeaacq GHsislcaaIXaGaaGilaiaaigdaaiaawUfacaGLDbaaaaa@3A0A@ (e.g., uniform kernel, Epanechnikov kernel), and when there are not at least two j s MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOAaiabgI Giolaadohaaaa@386A@ such that | x i x j | < λ . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaqWaaeaaca aMc8UaamiEamaaBaaaleaacaWGPbaabeaakiabgkHiTiaadIhadaWg aaWcbaGaamOAaaqabaGccaaMc8oacaGLhWUaayjcSdGaaGipaiabeU 7aSjaac6caaaa@4393@ To overcome this problem one can use a kernel function whose support is given by the whole real line (e.g., Gaussian kernel) or choose the bandwidth adaptively. The latter solution may also lead to more efficient estimators (see e.g., Fan and Gijbels 1992). With reference to the estimators F ^ * ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaja WaaWbaaSqabeaacaaIQaaaaOGaaGzaVpaabmaabaGaamiDaaGaayjk aiaawMcaaaaa@3AD1@ and F ˜ * ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaia WaaWbaaSqabeaacaaIQaaaaOGaaGzaVpaabmaabaGaamiDaaGaayjk aiaawMcaaaaa@3AD0@ based on the modified fitted values, it is moreover worth noting that one could in principle apply different bandwidths and/or regression weights to the y i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEamaaBa aaleaacaWGPbaabeaakiabgkHiTaaa@380E@ values and to the indicator functions. For the sake of simplicity, in the present work we shall consider neither adaptive bandwidth selection nor the possibility of different regression weights to estimate the mean regression function and the distributions of the error components.

Comparing the definitions of the estimators based on the two types of fitted values, it becomes immediately obvious that 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=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaia WaaeWaaeaacaWG0baacaGLOaGaayzkaaaaaa@385B@ are easier to compute since they are linear combinations of the observed indicator functions I ( y j t ) . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysamaabm aabaGaamyEamaaBaaaleaacaWGQbaabeaakiabgsMiJkaadshaaiaa wIcacaGLPaaacaGGUaaaaa@3CD9@ The coefficients of these linear combinations do not depend on the study variable Y MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamywaaaa@35DD@ and they can therefore be used to estimate averages of other functions than indicator functions, or of functions of several study variables, in particular when there are reasons to believe that the latter are related to the auxiliary variable X . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiwaiaac6 caaaa@368E@ This fact is of particular value to practitioners who want estimates related to several study variables to be consistent with one another. However, there is a strong argument in favor of the estimators F ^ * ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaja WaaWbaaSqabeaacaaIQaaaaOGaaGzaVpaabmaabaGaamiDaaGaayjk aiaawMcaaaaa@3AD1@ and F ˜ * ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaia WaaWbaaSqabeaacaaIQaaaaOGaaGzaVpaabmaabaGaamiDaaGaayjk aiaawMcaaaaa@3AD0@ based on the modified fitted values too: if y i = a + b x i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEamaaBa aaleaacaWGPbaabeaakiaai2dacaWGHbGaey4kaSIaamOyaiaadIha daWgaaWcbaGaamyAaaqabaaaaa@3CAE@ for all i U , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAaiabgI GiolaadwfacaGGSaaaaa@38FB@ then it follows that F ^ * ( t ) = F ˜ * ( t ) = F N ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaja WaaWbaaSqabeaacaaIQaaaaOGaaGzaVpaabmaabaGaamiDaaGaayjk aiaawMcaaiaai2daceWGgbGbaGaadaahaaWcbeqaaiaaiQcaaaGcca aMb8+aaeWaaeaacaWG0baacaGLOaGaayzkaaGaaGypaiaadAeadaWg aaWcbaGaamOtaaqabaGcdaqadaqaaiaadshaaiaawIcacaGLPaaaaa a@4686@ for every sample s MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4Caaaa@35F7@ such that the estimators are well-defined. One would therefore expect that F ^ * ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaja WaaWbaaSqabeaacaaIQaaaaOGaaGzaVpaabmaabaGaamiDaaGaayjk aiaawMcaaaaa@3AD1@ and F ˜ * ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaia WaaWbaaSqabeaacaaIQaaaaOGaaGzaVpaabmaabaGaamiDaaGaayjk aiaawMcaaaaa@3AD0@ be more efficient than 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=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaia WaaeWaaeaacaWG0baacaGLOaGaayzkaaaaaa@385B@ when there is a strong regression relationship between Y MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0de9 vqaqFeFr0xbba9Fa0P0RWFb9fq0hXdbbb9=e0dfrpm0dXdirVu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamywaaaa@35DD@ and X .

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