Régression quantile censurée pondérée

Section 3. Estimation des paramètres de la régression quantile censurée pondérée

Définissons la fonction de distribution de T i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamivamaaBa aaleaacaWGPbaabeaaaaa@37EA@ comme étant conditionnelle au vecteur de covariable de dimension p , X i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCaiaacY cacaaMe8UaaCiwamaaBaaaleaacaWGPbaabeaaaaa@3B24@ sous la forme F T i ( t | X i ) = Pr ( T i t | X i ) . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOramaaBa aaleaacaWGubWaaSbaaWqaaiaadMgaaeqaaaWcbeaakmaabmaabaGa amiDaiaaykW7daabbaqaaiaaykW7caWHybWaaSbaaSqaaiaadMgaae qaaaGccaGLhWoaaiaawIcacaGLPaaacqGH9aqpciGGqbGaaiOCamaa bmaabaGaamivamaaBaaaleaacaWGPbaabeaakiabgsMiJkaadshaca aMc8+aaqqaaeaacaaMc8UaaCiwamaaBaaaleaacaWGPbaabeaaaOGa ay5bSdaacaGLOaGaayzkaaGaaiOlaaaa@528F@ Soit Λ T i ( t | X i ) = log { 1 Pr ( T i t | X i ) } , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeu4MdW0aaS baaSqaaiaadsfadaWgaaadbaGaamyAaaqabaaaleqaaOWaaeWaaeaa caWG0bGaaGPaVpaaeeaabaGaaGPaVlaahIfadaWgaaWcbaGaamyAaa qabaaakiaawEa7aaGaayjkaiaawMcaaiabg2da9iabgkHiTiGacYga caGGVbGaai4zamaacmaabaGaaGPaVlaaigdacqGHsislciGGqbGaai OCamaabmaabaGaamivamaaBaaaleaacaWGPbaabeaakiabgsMiJkaa dshacaaMc8+aaqqaaeaacaaMc8UaaCiwamaaBaaaleaacaWGPbaabe aaaOGaay5bSdaacaGLOaGaayzkaaaacaGL7bGaayzFaaGaaiilaaaa @5C58@ N i ( t ) = I ( Y i t , δ i = 1 ) , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOtamaaBa aaleaacaWGPbaabeaakmaabmaabaGaamiDaaGaayjkaiaawMcaaiab g2da9mrr1ngBPrwtHrhAYaqeguuDJXwAKbstHrhAGq1DVbacfaGae8 hIWN0aaeWaaeaacaWGzbWaaSbaaSqaaiaadMgaaeqaaOGaeyizImQa amiDaiaacYcacaaMe8UaeqiTdq2aaSbaaSqaaiaadMgaaeqaaOGaey ypa0JaaGymaaGaayjkaiaawMcaaiaacYcaaaa@550F@ et M i ( t ) = N i ( t ) Λ T i ( t Λ Y i | X i ) . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamytamaaBa aaleaacaWGPbaabeaakmaabmaabaGaamiDaaGaayjkaiaawMcaaiab g2da9iaad6eadaWgaaWcbaGaamyAaaqabaGcdaqadaqaaiaadshaai aawIcacaGLPaaacqGHsislcqqHBoatdaWgaaWcbaGaamivamaaBaaa meaacaWGPbaabeaaaSqabaGcdaqadaqaaiaadshacqqHBoatcaWGzb WaaSbaaSqaaiaadMgaaeqaaOGaaGPaVpaaeeaabaGaaGPaVlaahIfa daWgaaWcbaGaamyAaaqabaaakiaawEa7aaGaayjkaiaawMcaaiaac6 caaaa@51DF@ Ainsi, Λ T i ( | X i ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeu4MdW0aaS baaSqaaiaadsfadaWgaaadbaGaamyAaaqabaaaleqaaOWaaeWaaeaa cqGHflY1caaMc8+aaqqaaeaacaaMc8UaaCiwamaaBaaaleaacaWGPb aabeaaaOGaay5bSdaacaGLOaGaayzkaaaaaa@4423@ est la fonction cumulative de risque conditionnelle à X i , N i ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCiwamaaBa aaleaacaWGPbaabeaakiaacYcacaaMe8UaamOtamaaBaaaleaacaWG PbaabeaakmaabmaabaGaamiDaaGaayjkaiaawMcaaaaa@3EB2@ est le processus de dénombrement et M i ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamytamaaBa aaleaacaWGPbaabeaakmaabmaabaGaamiDaaGaayjkaiaawMcaaaaa @3A6F@ est le processus des martingales associé à N i ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOtamaaBa aaleaacaWGPbaabeaakmaabmaabaGaamiDaaGaayjkaiaawMcaaaaa @3A70@ (Fleming et Harrington, 2011). Nous considérons un prolongement de la méthode d’estimation de la régression quantile censurée proposée par Peng et Huang (2008), intégrant les P i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiuamaaBa aaleaacaWGPbaabeaaaaa@37E6@ comme des poids connus découlant de l’information auxiliaire disponible au moyen du paramètre connu θ . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiUdeNaai Olaaaa@385F@ Il convient de souligner que θ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiUdehaaa@37AD@ et β ( τ ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCOSdmaabm aabaGaeqiXdqhacaGLOaGaayzkaaaaaa@3A83@ sont des paramètres distincts et que la fonction d’estimation g ( z : θ ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4zamaabm aabaGaamOEaiaaysW7caGG6aGaaGjbVlabeI7aXbGaayjkaiaawMca aaaa@3EF9@ utilisée pour calculer les P i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiuamaaBa aaleaacaWGPbaabeaaaaa@37E6@ est différente des fonctions d’estimation utilisées pour les paramètres de la régression quantile dans l’équation (1.1). Étant donné que les P i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiuamaaBa aaleaacaWGPbaabeaaaaa@37E6@ sont indépendants de β ( τ ) , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOSdi2aae WaaeaacqaHepaDaiaawIcacaGLPaaacaGGSaaaaa@3B96@ E { P i M i ( t ) | X i } = 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyramaacm aabaGaamiuamaaBaaaleaacaWGPbaabeaakiaad2eadaWgaaWcbaGa amyAaaqabaGcdaqadaqaaiaadshaaiaawIcacaGLPaaacaaMc8+aaq qaaeaacaaMc8UaaCiwamaaBaaaleaacaWGPbaabeaaaOGaay5bSdaa caGL7bGaayzFaaGaeyypa0JaaCimaaaa@47D1@ (par la propriété de martingale) pour t 0 , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiDaiabgw MiZkaaicdacaGGSaaaaa@3A20@ nous avons

E { n i = 1 n P i X i ( N i ( e X i β 0 ( τ ) ) Λ T [ e X i β 0 ( τ ) Y i | X i ] ) } = 0 , ( 3.1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyramaacm aabaWaaOaaaeaacaWGUbaaleqaaOWaaabCaeaacaWGqbWaaSbaaSqa aiaadMgaaeqaaOGaaCiwamaaBaaaleaacaWGPbaabeaakmaabmaaba GaamOtamaaBaaaleaacaWGPbaabeaakmaabmaabaGaamyzamaaCaaa leqabaGaaCiwamaaDaaameaacaWGPbaabaqefqvyO9wBHbacfaGaa8 hPdaaaliaahk7adaWgaaadbaGaaGimaaqabaWcdaqadaqaaiabes8a 0bGaayjkaiaawMcaaaaaaOGaayjkaiaawMcaaiabgkHiTiabfU5amn aaBaaaleaacaWGubaabeaakmaadmaabaWaaqGaaeaacaWGLbWaaWba aSqabeaacaWHybWaa0baaWqaaiaadMgaaeaacaWFKoaaaSGaaCOSdm aaBaaameaacaaIWaaabeaalmaabmaabaGaeqiXdqhacaGLOaGaayzk aaaaaOGaey4jIKTaamywamaaBaaaleaacaWGPbaabeaakiaaykW7ai aawIa7aiaaykW7caWHybWaaSbaaSqaaiaadMgaaeqaaaGccaGLBbGa ayzxaaaacaGLOaGaayzkaaaaleaacaWGPbGaeyypa0JaaGymaaqaai aad6gaa0GaeyyeIuoaaOGaay5Eaiaaw2haaiabg2da9iaahcdacaGG SaGaaGzbVlaaywW7caaMf8UaaiikaiaaiodacaGGUaGaaGymaiaacM caaaa@7979@

β 0 ( τ ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCOSdmaaBa aaleaacaaIWaaabeaakmaabmaabaGaeqiXdqhacaGLOaGaayzkaaaa aa@3B73@ est la valeur réelle de β ( τ ) , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCOSdmaabm aabaGaeqiXdqhacaGLOaGaayzkaaGaaiilaaaa@3B33@ dans l’équation (1.2) pour un quantile donné, τ . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiXdqNaai Olaaaa@386E@

Étant donné que Λ T i ( | X i ) , i = 1 , 2 , , n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeu4MdW0aaS baaSqaaiaadsfadaWgaaadbaGaamyAaaqabaaaleqaaOWaaeWaaeaa cqGHflY1caaMc8+aaqqaaeaacaaMc8UaaCiwamaaBaaaleaacaWGPb aabeaaaOGaay5bSdaacaGLOaGaayzkaaGaaiilaiaaysW7caWGPbGa eyypa0JaaGymaiaacYcacaaMe8UaaGOmaiaacYcacaaMe8UaeSOjGS KaaiilaiaaysW7caWGUbaaaa@5297@ sont des fonctions inconnues, Peng et Huang (2008) ont calculé la relation entre Λ T [ e X i β 0 ( τ ) Y i | X i ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeu4MdW0aaS baaSqaaiaadsfaaeqaaOWaamWaaeaacaWGLbWaaWbaaSqabeaacaWH ybWaa0baaWqaaiaadMgaaeaaruavHH2BTfgaiuaacaWFKoaaaSGaaC OSdmaaBaaameaacaaIWaaabeaalmaabmaabaGaeqiXdqhacaGLOaGa ayzkaaaaaOGaey4jIKTaamywamaaBaaaleaacaWGPbaabeaakiaayk W7daabbaqaaiaaykW7caWHybWaaSbaaSqaaiaadMgaaeqaaaGccaGL hWoaaiaawUfacaGLDbaaaaa@5118@ et β 0 ( τ ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCOSdmaaBa aaleaacaaIWaaabeaakmaabmaabaGaeqiXdqhacaGLOaGaayzkaaaa aa@3B73@ pour utiliser l’équation (3.1) afin d’estimer la valeur de β 0 ( τ ) . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCOSdmaaBa aaleaacaaIWaaabeaakmaabmaabaGaeqiXdqhacaGLOaGaayzkaaGa aiOlaaaa@3C25@ Compte tenu que F F [ e X i β 0 ( u ) | X i ] = τ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOramaaBa aaleaatuuDJXwAKzKCHTgD1jharyqr1ngBPrgigjxyRrxDYbacfaGa e8xbWBeabeaakmaadmaabaGaamyzamaaCaaaleqabaGaaCiwamaaDa aameaacaWGPbaabaqefqvyO9wBHbacgaGaa4hPdaaaliaahk7adaWg aaadbaGaaGimaaqabaWcdaqadaqaaiaadwhaaiaawIcacaGLPaaaaa GccaaMc8+aaqqaaeaacaaMc8UaaCiwamaaBaaaleaacaWGPbaabeaa aOGaay5bSdaacaGLBbGaayzxaaGaeyypa0JaeqiXdqhaaa@5A66@ et en utilisant la monotonicité de X i T β 0 ( τ ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiwamaaDa aaleaacaWGPbaabaGaamivaaaakiabek7aInaaBaaaleaacaaIWaaa beaakmaabmaabaGaeqiXdqhacaGLOaGaayzkaaaaaa@3EB1@ dans τ , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiXdqNaai ilaaaa@386C@ ils ont démontré que Λ T [ e X i β 0 ( τ ) Y i | X i ] = 0 τ I [ Y i e X i β ( u ) ] d H ( u ) , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeu4MdW0aaS baaSqaaiaadsfaaeqaaOWaamWaaeaacaWGLbWaaWbaaSqabeaacaWH ybWaa0baaWqaaiaadMgaaeaaruavHH2BTfgaiuaacaWFKoaaaSGaaC OSdmaaBaaameaacaaIWaaabeaalmaabmaabaGaeqiXdqhacaGLOaGa ayzkaaaaaOGaey4jIKTaamywamaaBaaaleaacaWGPbaabeaakiaayk W7daabbaqaaiaaykW7caWHybWaaSbaaSqaaiaadMgaaeqaaaGccaGL hWoaaiaawUfacaGLDbaacqGH9aqpdaWdXaqaamrr1ngBPrwtHrhAYa qeguuDJXwAKbstHrhAGq1DVbacgaGae4hIWN0aamWaaeaacaWGzbWa aSbaaSqaaiaadMgaaeqaaOGaeyyzImRaamyzamaaCaaaleqabaGaaC iwamaaDaaameaacaWGPbaabaGaa8hPdaaaliaahk7adaqadaqaaiaa dwhaaiaawIcacaGLPaaaaaaakiaawUfacaGLDbaacaWGKbGaamisam aabmaabaGaamyDaaGaayjkaiaawMcaaiaacYcaaSqaaiaaicdaaeaa cqaHepaDa0Gaey4kIipaaaa@7564@ H ( u ) = log ( 1 u ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamisamaabm aabaGaamyDaaGaayjkaiaawMcaaiabg2da9iabgkHiTiGacYgacaGG VbGaai4zamaabmaabaGaaGymaiabgkHiTiaadwhaaiaawIcacaGLPa aaaaa@4235@ pour 0 u < 1. MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGimaiabgs MiJkaadwhacqGH8aapcaaIXaGaaiOlaaaa@3BD1@

Notre équation d’estimation de la régression quantile censurée pondérée est donc

n S n ( β , τ ) = 0 , ( 3.2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOaaaeaaca WGUbaaleqaaOGaaGjcVlaadofadaWgaaWcbaGaamOBaaqabaGcdaqa daqaaiaahk7acaGGSaGaaGjbVlabes8a0bGaayjkaiaawMcaaiabg2 da9iaahcdacaGGSaGaaGzbVlaaywW7caaMf8UaaGzbVlaaywW7caGG OaGaaG4maiaac6cacaaIYaGaaiykaaaa@4F22@

S n ( β , τ ) = i = 1 n P i X i { i ( e X i β ( τ ) ) 0 τ I [ Y i e X i β ( u ) ] d H ( u ) } . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uamaaBa aaleaacaWGUbaabeaakmaabmaabaGaaCOSdiaacYcacaaMe8UaeqiX dqhacaGLOaGaayzkaaGaeyypa0ZaaabCaeaacaWGqbWaaSbaaSqaai aadMgaaeqaaOGaaCiwamaaBaaaleaacaWGPbaabeaakmaacmaabaWe fv3ySLgznfgDOjdaryqr1ngBPrginfgDObcv39gaiuqacqWFveItda WgaaWcbaGaamyAaaqabaGcdaqadaqaaiaadwgadaahaaWcbeqaaiaa hIfadaqhaaadbaGaamyAaaqaaerbufgAV1wyaGGbaiaa+r6aaaWcca WHYoWaaeWaaeaacqaHepaDaiaawIcacaGLPaaaaaaakiaawIcacaGL PaaacqGHsisldaWdXaqaaGqbaiab9Hi8jnaadmaabaGaamywamaaBa aaleaacaWGPbaabeaakiabgwMiZkaadwgadaahaaWcbeqaaiaahIfa daqhaaadbaGaamyAaaqaaiaa+r6aaaWccaWHYoWaaeWaaeaacaWG1b aacaGLOaGaayzkaaaaaaGccaGLBbGaayzxaaGaamizaiaadIeadaqa daqaaiaadwhaaiaawIcacaGLPaaaaSqaaiaaicdaaeaacqaHepaDa0 Gaey4kIipaaOGaay5Eaiaaw2haaiaac6caaSqaaiaadMgacqGH9aqp caaIXaaabaGaamOBaaqdcqGHris5aaaa@7F33@

Ici, les P i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiuamaaBa aaleaacaWGPbaabeaaaaa@37E6@ sont définis dans l’équation (2.2). Soit s ( β , τ ) = E { S n ( β , τ ) } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4Camaabm aabaGaaCOSdiaacYcacaaMe8UaeqiXdqhacaGLOaGaayzkaaGaeyyp a0JaamyramaacmaabaGaam4uamaaBaaaleaacaWGUbaabeaakmaabm aabaGaaCOSdiaacYcacaaMe8UaeqiXdqhacaGLOaGaayzkaaaacaGL 7bGaayzFaaaaaa@4A83@ et la propriété de martingale de M ( ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWefv3ySLgznf gDOjdaryqr1ngBPrginfgDObcv39gaiuqacqWFmcFtdaqadaqaaiab gwSixdGaayjkaiaawMcaaaaa@45BC@ donne s ( β 0 , τ ) = 0 . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4Camaabm aabaGaaCOSdmaaBaaaleaacaaIWaaabeaakiaacYcacaaMe8UaeqiX dqhacaGLOaGaayzkaaGaeyypa0JaaCimaiaac6caaaa@4119@ Pour un quantile en particulier, τ k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiXdq3aaS baaSqaaiaadUgaaeqaaaaa@38D8@ et un estimateur de β 0 ( τ k ) , β ^ ( τ k ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCOSdmaaBa aaleaacaaIWaaabeaakmaabmaabaGaeqiXdq3aaSbaaSqaaiaadUga aeqaaaGccaGLOaGaayzkaaGaaiilaiaaysW7ceWHYoGbaKaadaqada qaaiabes8a0naaBaaaleaacaWGRbaabeaaaOGaayjkaiaawMcaaaaa @4498@ est une fonction en escaliers continue à droite qui fait des sauts seulement selon la définition d’une grille, S L = { 0 = τ 0 < τ 1 < < τ L = τ U < 1 } . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWefv3ySLgznf gDOjdaryqr1ngBPrginfgDObcv39gaiuqacqWFsc=udaWgaaWcbaGa amitaaqabaGccqGH9aqpdaGadaqaaiaaicdacqGH9aqpcqaHepaDda WgaaWcbaGaaGimaaqabaGccqGH8aapcqaHepaDdaWgaaWcbaGaaGym aaqabaGccqGH8aapcqWIMaYscqGH8aapcqaHepaDdaWgaaWcbaGaam itaaqabaGccqGH9aqpcqaHepaDdaWgaaWcbaGaamyvaaqabaGccqGH 8aapcaaIXaaacaGL7bGaayzFaaGaaiOlaaaa@5AA4@ Ici, L MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamitaaaa@36C8@ dépend de la taille de l’échantillon, n . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaiaac6 caaaa@379C@ La taille de S L MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWefv3ySLgznf gDOjdaryqr1ngBPrginfgDObcv39gaiuqacqWFsc=udaWgaaWcbaGa amitaaqabaaaaa@42F2@ est définie comme étant S L = max k ( τ k τ k 1 ) . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaauWaaeaaca aMi8+efv3ySLgznfgDOjdaryqr1ngBPrginfgDObcv39gaiuqacqWF sc=udaWgaaWcbaGaamitaaqabaGccaaMi8oacaGLjWUaayPcSdGaey ypa0ZaaCbeaeaaciGGTbGaaiyyaiaacIhaaSqaaiaadUgaaeqaaOWa aeWaaeaacqaHepaDdaWgaaWcbaGaam4AaaqabaGccqGHsislcqaHep aDdaWgaaWcbaGaam4AaiabgkHiTiaaigdaaeqaaaGccaGLOaGaayzk aaGaaiOlaaaa@58F8@

Pour des quantiles différents, τ 0 , τ 1 , , τ L ( 0 = τ 0 < τ 1 < < τ L < 1 ) , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiXdq3aaS baaSqaaiaaicdaaeqaaOGaaiilaiaaysW7cqaHepaDdaWgaaWcbaGa aGymaaqabaGccaGGSaGaaGjbVlablAciljaacYcacaaMe8UaeqiXdq 3aaSbaaSqaaiaadYeaaeqaaOWaaeWaaeaacaaIWaGaeyypa0JaeqiX dq3aaSbaaSqaaiaaicdaaeqaaOGaeyipaWJaeqiXdq3aaSbaaSqaai aaigdaaeqaaOGaeyipaWJaeSOjGSKaeyipaWJaeqiXdq3aaSbaaSqa aiaadYeaaeqaaOGaeyipaWJaaGymaaGaayjkaiaawMcaaiaacYcaaa a@5824@ d’après l’équation (3.2), nous pouvons obtenir β ^ ( τ k ) ( k = 1 , 2 , , L ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabCOSdyaaja WaaeWaaeaacqaHepaDdaWgaaWcbaGaam4AaaqabaaakiaawIcacaGL PaaadaqadaqaaiaadUgacqGH9aqpcaaIXaGaaiilaiaaysW7caaIYa GaaiilaiaaysW7cqWIMaYscaGGSaGaaGjbVlaadYeaaiaawIcacaGL Paaaaaa@4959@ en résolvant de manière récursive l’équation d’estimation monotone suivante pour β ( τ k ) : MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCOSdmaabm aabaGaeqiXdq3aaSbaaSqaaiaadUgaaeqaaaGccaGLOaGaayzkaaGa aGjbVlaacQdaaaa@3DF4@

n i = 1 n P i X i { i ( e X i β ( τ k ) ) r = 0 k 1 I [ Y i e X i β ^ ( τ r ) ] { H ( τ r + 1 ) H ( τ r ) } } = 0 . ( 3.3 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOaaaeaaca WGUbaaleqaaOWaaabCaeaacaWGqbWaaSbaaSqaaiaadMgaaeqaaOGa aCiwamaaBaaaleaacaWGPbaabeaaaeaacaWGPbGaeyypa0JaaGymaa qaaiaad6gaa0GaeyyeIuoakmaacmaabaWefv3ySLgznfgDOjdaryqr 1ngBPrginfgDObcv39gaiuqacqWFveItdaWgaaWcbaGaamyAaaqaba GcdaqadaqaaiaadwgadaahaaWcbeqaaiaahIfadaqhaaadbaGaamyA aaqaaerbufgAV1wyaGGbaiaa+r6aaaWccaWHYoWaaeWaaeaacqaHep aDdaWgaaadbaGaam4AaaqabaaaliaawIcacaGLPaaaaaaakiaawIca caGLPaaacqGHsisldaaeWbqaaGqbaiab9Hi8jnaadmaabaGaamywam aaBaaaleaacaWGPbaabeaakiabgwMiZkaadwgadaahaaWcbeqaaiaa hIfadaqhaaadbaGaamyAaaqaaiaa+r6aaaWcceWHYoGbaKaadaqada qaaiabes8a0naaBaaameaacaWGYbaabeaaaSGaayjkaiaawMcaaaaa aOGaay5waiaaw2faaaWcbaGaamOCaiabg2da9iaaicdaaeaacaWGRb GaeyOeI0IaaGymaaqdcqGHris5aOWaaiWaaeaacaWGibWaaeWaaeaa cqaHepaDdaWgaaWcbaGaamOCaiabgUcaRiaaigdaaeqaaaGccaGLOa GaayzkaaGaeyOeI0IaamisamaabmaabaGaeqiXdq3aaSbaaSqaaiaa dkhaaeqaaaGccaGLOaGaayzkaaaacaGL7bGaayzFaaaacaGL7bGaay zFaaGaeyypa0JaaCimaiaac6cacaaMf8UaaGzbVlaaywW7caGGOaGa aG4maiaac6cacaaIZaGaaiykaaaa@918B@

Nous définissons les estimateurs, β ^ ( τ k ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabCOSdyaaja WaaeWaaeaacqaHepaDdaWgaaWcbaGaam4AaaqabaaakiaawIcacaGL Paaaaaa@3BB9@ comme les solutions généralisées (Fygenson et Ritov, 1994), parce que l’équation (3.3) n’est pas continue et parce que la solution peut ne pas être unique.

3.1  Théorie asymptotique

Nous avons calculé les propriétés asymptotiques des estimateurs par régression quantile censurée pondérée fondée sur la VE en utilisant l’approche de Peng et Huang (2008). Maintenant, nous démontrons la cohérence uniforme et la faible convergence gaussienne de l’estimateur par régression quantile censurée pondérée proposée, β ^ ( ) . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabCOSdyaaja WaaeWaaeaacqGHflY1aiaawIcacaGLPaaacaGGUaaaaa@3BCA@

Définissons F ( t | X ) = Pr ( Y t | X ) , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOramaabm aabaGaamiDaiaaykW7daabbaqaaiaaykW7caWHybaacaGLhWoaaiaa wIcacaGLPaaacqGH9aqpciGGqbGaaiOCamaabmaabaGaamywaiabgs MiJkaadshacaaMc8+aaqqaaeaacaaMc8UaaCiwaaGaay5bSdaacaGL OaGaayzkaaGaaiilaaaa@4CF1@ F ¯ ( t | X ) = Pr ( Y > t | X ) , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaara WaaeWaaeaacaWG0bGaaGPaVpaaeeaabaGaaGPaVlaahIfaaiaawEa7 aaGaayjkaiaawMcaaiabg2da9iGaccfacaGGYbWaaeWaaeaacaWGzb GaeyOpa4JaamiDaiaaykW7daabbaqaaiaaykW7caWHybaacaGLhWoa aiaawIcacaGLPaaacaGGSaaaaa@4C5C@ F ˜ ( t | X ) = Pr ( Y t , δ = 1 | X ) , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaia WaaeWaaeaacaWG0bGaaGPaVpaaeeaabaGaaGPaVlaahIfaaiaawEa7 aaGaayjkaiaawMcaaiabg2da9iGaccfacaGGYbWaaeWaaeaacaWGzb GaeyizImQaamiDaiaacYcacaaMe8UaeqiTdqMaeyypa0JaaGymaiaa ykW7daabbaqaaiaaykW7caWHybaacaGLhWoaaiaawIcacaGLPaaaca GGSaaaaa@52A3@ f ¯ ( y | X ) = f ( y | X ) = d F ( y | X ) / d y MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOzayaara WaaeWaaeaacaWG5bGaaGPaVpaaeeaabaGaaGPaVlaahIfaaiaawEa7 aaGaayjkaiaawMcaaiabg2da9iabgkHiTiaadAgadaqadaqaaiaadM hacaaMc8+aaqqaaeaacaaMc8UaaCiwaaGaay5bSdaacaGLOaGaayzk aaGaeyypa0ZaaSGbaeaacqGHsislcaWGKbGaamOramaabmaabaGaam yEaiaaykW7daabbaqaaiaaykW7caWHybaacaGLhWoaaiaawIcacaGL PaaaaeaacaWGKbGaamyEaaaaaaa@57B2@ et f ˜ ( y | X ) = d F ˜ ( y | X ) / d y . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSGbaeaace WGMbGbaGaadaqadaqaaiaadMhacaaMc8+aaqqaaeaacaaMc8UaaCiw aaGaay5bSdaacaGLOaGaayzkaaGaeyypa0JaamizaiqadAeagaacam aabmaabaGaamyEaiaaykW7daabbaqaaiaaykW7caWHybaacaGLhWoa aiaawIcacaGLPaaaaeaacaWGKbGaamyEaaaacaGGUaaaaa@4C8D@ (Pour un vecteur h , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiAaiaacY caaaa@3794@ h 2 = h h T , h ( l ) = l e MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiAamaaCa aaleqabaGaey4LIqSaaGOmaaaakiabg2da9iaadIgacaWGObWaaWba aSqabeaacaWGubaaaOGaaGzaVlaacYcacaaMe8UaamiAamaaCaaale qabaWaaeWaaeaacaWGSbaacaGLOaGaayzkaaaaaOGaeyypa0JaamiB amaaCaaaleqabaGaaeyzaaaaaaa@4841@ comme composante de h , h MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiAaiaacY cacaaMe8+aauWaaeaacaaMc8UaamiAaiaaykW7aiaawMa7caGLkWoa aaa@404B@ est la norme euclidienne de h . ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiAaiaac6 cacaGGPaaaaa@3843@

Définissons W i = λ θ 0 g ( Z i ; θ 0 ) X i , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaC4vamaaBa aaleaacaWGPbaabeaakiabg2da9iabeU7aSnaaDaaaleaacqaH4oqC daWgaaadbaGaaGimaaqabaaaleaaruavHH2BTfgaiuaacaWFKoaaaO Gaam4zamaabmaabaGaaCOwamaaBaaaleaacaWGPbaabeaakiaacUda caaMe8UaaCiUdmaaBaaaleaacaaIWaaabeaaaOGaayjkaiaawMcaai aahIfadaWgaaWcbaGaamyAaaqabaGccaGGSaaaaa@4CEA@ i = 1 , 2 , , n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAaiabg2 da9iaaigdacaGGSaGaaGjbVlaaikdacaGGSaGaaGjbVlablAciljaa cYcacaaMe8UaamOBaaaa@422E@ en tant que vecteur de dimension p . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCaiaac6 caaaa@379E@

Conditions de régularité :

  • R.1 :
  • Les observations, Z i , i = 1 , 2 , , n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCOwamaaBa aaleaacaWGPbaabeaakiaacYcacaaMe8UaamyAaiabg2da9iaaigda caGGSaGaaGjbVlaaikdacaGGSaGaaGjbVlablAciljaacYcacaaMe8 UaamOBaaaa@4672@ sont des observations i.i.d. issues d’une certaine distribution. Sans perte de généralité, nous supposons que ( Y i , δ i , X i ) Z i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaaca WGzbWaaSbaaSqaaiaadMgaaeqaaOGaaiilaiaaysW7cqaH0oazdaWg aaWcbaGaamyAaaqabaGccaGGSaGaaGjbVlaahIfadaqhaaWcbaGaam yAaaqaaerbufgAV1wyaGqbaiaa=r6aaaaakiaawIcacaGLPaaadaah aaWcbeqaaiaa=r6aaaGccqGHckcZcaWHAbWaaSbaaSqaaiaadMgaae qaaaaa@4BC8@ pour tous les i = 1 , 2 , , n . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAaiabg2 da9iaaigdacaGGSaGaaGjbVlaaikdacaGGSaGaaGjbVlablAciljaa cYcacaaMe8UaamOBaiaac6caaaa@42E0@
  • R.2 :
  • Il existe θ 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCiUdmaaBa aaleaacaaIWaaabeaaaaa@3820@ tel que E { g ( Z i ; θ 0 ) } = 0 , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyramaacm aabaGaam4zamaabmaabaGaaCOwamaaBaaaleaacaWGPbaabeaakiaa cUdacaaMe8UaaCiUdmaaBaaaleaacaaIWaaabeaaaOGaayjkaiaawM caaaGaay5Eaiaaw2haaiabg2da9iaaicdacaGGSaaaaa@445E@ la matrice Σ ( θ 0 ) = E { g ( Z i ; θ 0 ) g ( Z i ; θ 0 ) } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaC4Odmaabm aabaGaaCiUdmaaBaaaleaacaaIWaaabeaaaOGaayjkaiaawMcaaiab g2da9iaadweadaGadaqaaiaadEgadaqadaqaaiaahQfadaWgaaWcba GaamyAaaqabaGccaGG7aGaaGjbVlaahI7adaWgaaWcbaGaaGimaaqa baaakiaawIcacaGLPaaacaWGNbWaaeWaaeaacaWHAbWaaSbaaSqaai aadMgaaeqaaOGaai4oaiaaysW7caWH4oWaaSbaaSqaaiaaicdaaeqa aaGccaGLOaGaayzkaaWaaWbaaSqabeaaruavHH2BTfgaiuaacaWFKo aaaaGccaGL7bGaayzFaaaaaa@54B7@ est une matrice définie positive, g ( z ; θ ) θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSqaaSqaai abgkGi2kaadEgadaqadaqaaiaahQhacaGG7aGaaGjbVlaahI7aaiaa wIcacaGLPaaaaeaacqGHciITcaWH4oaaaaaa@412A@ est continue dans le voisinage de θ 0 . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCiUdmaaBa aaleaacaaIWaaabeaakiaac6caaaa@38DC@ La matrice E { g ( Z ; θ ) θ } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyramaacm aabaWaaSqaaSqaaiabgkGi2kaadEgadaqadaqaaiaahQfacaGG7aGa aGjbVlaahI7aaiaawIcacaGLPaaaaeaacqGHciITcaWH4oaaaaGcca GL7bGaayzFaaaaaa@440F@ est de plein rang.
  • R.3 :
  • Il existe les fonctions H l j ( z ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamisamaaBa aaleaacaWGSbGaamOAaaqabaGcdaqadaqaaiaahQhaaiaawIcacaGL Paaaaaa@3B65@ tel que pour θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCiUdaaa@373A@ dans le voisinage de θ 0 , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCiUdmaaBa aaleaacaaIWaaabeaakiaacYcaaaa@38DA@ | g l ( z ; θ ) θ j | H l j ( z ) , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaqWaaeaaca aMc8+aaSqaaSqaaiabgkGi2kaadEgadaWgaaadbaGaamiBaaqabaWc daqadaqaaiaahQhacaGG7aGaaGjbVlaahI7aaiaawIcacaGLPaaaae aacqGHciITcqaH4oqCdaWgaaadbaGaamOAaaqabaaaaOGaaGPaVdGa ay5bSlaawIa7aiabgsMiJkaadIeadaWgaaWcbaGaamiBaiaadQgaae qaaOWaaeWaaeaacaWH6baacaGLOaGaayzkaaGaaiilaaaa@51F7@ où pour une constante C , E { H l j 2 ( Z ) } C < MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4qaiaacY cacaaMe8UaamyramaacmaabaGaamisamaaDaaaleaacaWGSbGaamOA aaqaaiaaikdaaaGcdaqadaqaaiaahQfaaiaawIcacaGLPaaaaiaawU hacaGL9baacqGHKjYOcaWGdbGaeyipaWJaeyOhIukaaa@46F4@ pour l = 1 , , q MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiBaiabg2 da9iaaigdacaGGSaGaaGjbVlablAciljaacYcacaaMe8UaamyCaaaa @3F3A@ et j = 1 , , d . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOAaiabg2 da9iaaigdacaGGSaGaaGjbVlablAciljaacYcacaaMe8Uaamizaiaa c6caaaa@3FDD@
  • R.4 :
  • sup i X i < MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaCbeaeaaci GGZbGaaiyDaiaacchaaSqaaiaadMgaaeqaaOWaauWaaeaacaaMc8Ua aCiwamaaBaaaleaacaWGPbaabeaakiaaykW7aiaawMa7caGLkWoacq GH8aapcqGHEisPaaa@44C4@ et sup i X i Y i < . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaCbeaeaaci GGZbGaaiyDaiaacchaaSqaaiaadMgaaeqaaOWaauWaaeaacaaMc8Ua aCiwamaaBaaaleaacaWGPbaabeaakiaahMfadaWgaaWcbaGaamyAaa qabaGccaaMc8oacaGLjWUaayPcSdGaeyipaWJaeyOhIuQaaiOlaaaa @477C@
  • R.5 :
  1. Chaque composante de E [ X ( e X β 0 ( τ ) ) ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyramaadm aabaGaaCiwamrr1ngBPrwtHrhAYaqeguuDJXwAKbstHrhAGq1DVbac feGae8xfH40aaeWaaeaacaWGLbWaaWbaaSqabeaacaWHybWaaWbaaW qabeaaruavHH2BTfgaiyaacaGFKoaaaSGaaCOSdmaaBaaameaacaaI WaaabeaalmaabmaabaGaeqiXdqhacaGLOaGaayzkaaaaaaGccaGLOa GaayzkaaaacaGLBbGaayzxaaaaaa@5127@ est une fonction Lipschitz de τ . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiXdqNaai Olaaaa@386D@
  2. f ˜ ( t | x ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOzayaaia WaaeWaaeaacaWG0bGaaGPaVpaaeeaabaGaaGPaVlaahIhaaiaawEa7 aaGaayjkaiaawMcaaaaa@3F1D@ et f ( t | x ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaabm aabaGaamiDaiaaykW7daabbaqaaiaaykW7caWH4baacaGLhWoaaiaa wIcacaGLPaaaaaa@3F0E@ sont bornées supérieurement uniformément en t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiDaaaa@36EF@ et x . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCiEaiaac6 caaaa@37A9@
  • R.6 :
  1. f ˜ ( e X b | X ) > 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOzayaaia WaaeWaaeaacaWGLbWaaWbaaSqabeaacaWHybWaaWbaaWqabeaaruav HH2BTfgaiuaacaWFKoaaaSGaaCOyaaaakiaaykW7daabbaqaaiaayk W7caWHybaacaGLhWoaaiaawIcacaGLPaaacqGH+aGpcaaIWaaaaa@4691@ pour toutes les b B ( d 0 ) , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCOyaiabgI Gioprr1ngBPrMrYf2A0vNCaeHbfv3ySLgzGyKCHTgD1jhaiuaacqWF baVqdaqadaqaaiaadsgadaWgaaWcbaGaaGimaaqabaaakiaawIcaca GLPaaacaGGSaaaaa@48F0@ B ( d ) = { b p : inf τ ( 0 , τ U ) μ ( b ) μ { β 0 ( τ ) } d } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWefv3ySLgzgj xyRrxDYbqeguuDJXwAKbIrYf2A0vNCaGqbaiab=fa8cnaabmaabaGa amizaaGaayjkaiaawMcaaiabg2da9maacmaabaGaaCOyaiabgIGiol ab=XrisnaaCaaaleqabaGaamiCaaaakiaaysW7caGG6aGaaGjbVpaa xababaGaciyAaiaac6gacaGGMbaaleaacqaHepaDcqGHiiIZdaqada qaaiaaicdacaGGSaGaaGjbVlabes8a0naaBaaameaacaWGvbaabeaa aSGaayjkaiaawMcaaaqabaGcdaqbdaqaaiaaykW7caWH8oWaaeWaae aacaWHIbaacaGLOaGaayzkaaGaeyOeI0IaaCiVdmaacmaabaGaaCOS dmaaBaaaleaacaaIWaaabeaakmaabmaabaGaeqiXdqhacaGLOaGaay zkaaaacaGL7bGaayzFaaGaaGPaVdGaayzcSlaawQa7aiabgsMiJkaa dsgaaiaawUhacaGL9baaaaa@74C6@ pour d > 0 , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamizaiabg6 da+iaaicdacaGGSaaaaa@3951@ et μ ( b ) = E [ X ( e X b ) ] , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCiVdmaabm aabaGaaCOyaaGaayjkaiaawMcaaiabg2da9iaadweadaWadaqaaiaa hIfatuuDJXwAK1uy0HMmaeHbfv3ySLgzG0uy0HgiuD3BaGqbbiab=v rionaabmaabaGaamyzamaaCaaaleqabaGaaCiwamaaCaaameqabaqe fqvyO9wBHbacgaGaa4hPdaaaliaahkgaaaaakiaawIcacaGLPaaaai aawUfacaGLDbaacaGGSaaaaa@5206@ est un voisinage contenant { β 0 ( τ ) , τ ( 0 , τ U ) } . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiWaaeaaca WHYoWaaSbaaSqaaiaaicdaaeqaaOWaaeWaaeaacqaHepaDaiaawIca caGLPaaacaGGSaGaaGjbVlabes8a0jabgIGiopaabmaabaGaaGimai aacYcacaaMe8UaeqiXdq3aaSbaaSqaaiaadwfaaeqaaaGccaGLOaGa ayzkaaaacaGL7bGaayzFaaGaaiOlaaaa@4B30@
  2. Pour avoir une matrice définie positive, E { X 2 } > 0. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyramaacm aabaGaaCiwamaaCaaaleqabaGaey4LIqSaaGOmaaaaaOGaay5Eaiaa w2haaiabg6da+iaaicdacaGGUaaaaa@3F42@
  3. Chaque composante de E [ X 2 f ¯ ( e X b | X ) e X b ] × ( E [ X 2 f ˜ ( e X b | X ) e X b ] ) 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyramaadm aabaGaaCiwamaaCaaaleqabaGaey4LIqSaaGOmaaaakiqadAgagaqe amaabmaabaWaaqGaaeaacaWGLbWaaWbaaSqabeaacaWHybWaaWbaaW qabeaaruavHH2BTfgaiuaacaWFKoaaaSGaaCOyaaaakiaaykW7aiaa wIa7aiaaykW7caWHybaacaGLOaGaayzkaaGaamyzamaaCaaaleqaba GaaCiwamaaCaaameqabaGaa8hPdaaaliaahkgaaaaakiaawUfacaGL DbaacqGHxdaTdaqadaqaaiaadweadaWadaqaaiaahIfadaahaaWcbe qaaiabgEPielaaikdaaaGcceWGMbGbaGaadaqadaqaamaaeiaabaGa amyzamaaCaaaleqabaGaaCiwamaaCaaameqabaGaa8hPdaaaliaahk gaaaGccaaMc8oacaGLiWoacaaMc8UaaCiwaaGaayjkaiaawMcaaiaa dwgadaahaaWcbeqaaiaahIfadaahaaadbeqaaiaa=r6aaaWccaWHIb aaaaGccaGLBbGaayzxaaaacaGLOaGaayzkaaWaaWbaaSqabeaacqGH sislcaaIXaaaaaaa@6C7B@ est bornée uniformément dans b B ( d 0 ) ; B ( d 0 ) . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCOyaiabgI Gioprr1ngBPrMrYf2A0vNCaeHbfv3ySLgzGyKCHTgD1jhaiuaacqWF baVqdaqadaqaaiaadsgadaWgaaWcbaGaaGimaaqabaaakiaawIcaca GLPaaacaGG7aGaaGjbVlab=fa8cnaabmaabaGaamizamaaBaaaleaa caaIWaaabeaaaOGaayjkaiaawMcaaiaac6caaaa@5058@
  • R.7 :
  • Pour chaque v ( 0 , τ U ) , inf τ [ v , τ U ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODaiabgI GiopaabmaabaGaaGimaiaacYcacaaMe8UaeqiXdq3aaSbaaSqaaiaa dwfaaeqaaaGccaGLOaGaayzkaaGaaiilamaaxababaGaciyAaiaac6 gacaGGMbaaleaacqaHepaDcqGHiiIZdaWadaqaaiaadAhacaGGSaGa aGjbVlabes8a0naaBaaameaacaWGvbaabeaaaSGaay5waiaaw2faaa qabaaaaa@4EC8@ eigmin  E [ X 2 f ˜ ( e X β 0 ( τ ) | X ) e X β 0 ( τ ) ] > 0 , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyramaadm aabaGaaCiwamaaCaaaleqabaGaey4LIqSaaGOmaaaakiqadAgagaac amaabmaabaWaaqGaaeaacaWGLbWaaWbaaSqabeaacaWHybWaaWbaaW qabeaaruavHH2BTfgaiuaacaWFKoaaaSGaaCOSdmaaBaaameaacaaI WaaabeaalmaabmaabaGaeqiXdqhacaGLOaGaayzkaaaaaOGaaGPaVd GaayjcSdGaaGPaVlaahIfaaiaawIcacaGLPaaacaWGLbWaaWbaaSqa beaacaWHybWaaWbaaWqabeaacaWFKoaaaSGaaCOSdmaaBaaameaaca aIWaaabeaalmaabmaabaGaeqiXdqhacaGLOaGaayzkaaaaaaGccaGL BbGaayzxaaGaeyOpa4JaaGimaiaacYcaaaa@5B50@ où eigmin ( ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaacq GHflY1aiaawIcacaGLPaaaaaa@39C9@ est la valeur propre minimale d’une matrice.

Théorème 1. En supposant que les conditions de régularité R.1 à R.7 soient vérifiées, si lim n S L = 0 , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaCbeaeaaci GGSbGaaiyAaiaac2gaaSqaaiaad6gacqGHsgIRcqGHEisPaeqaaOWa auWaaeaacaaMc8+efv3ySLgznfgDOjdaryqr1ngBPrginfgDObcv39 gaiuqacqWFsc=udaWgaaWcbaGaamitaaqabaGccaaMc8oacaGLjWUa ayPcSdGaeyypa0JaaGimaiaacYcaaaa@530C@ alors sup τ [ v , τ U ] β ^ ( τ ) β 0 ( τ ) p 0 , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaCbeaeaaci GGZbGaaiyDaiaacchaaSqaaiabes8a0jabgIGiopaadmaabaGaamOD aiaacYcacaaMe8UaeqiXdq3aaSbaaWqaaiaadwfaaeqaaaWccaGLBb GaayzxaaaabeaakmaafmaabaGaaGPaVlqahk7agaqcamaabmaabaGa eqiXdqhacaGLOaGaayzkaaGaeyOeI0IaaCOSdmaaBaaaleaacaaIWa aabeaakmaabmaabaGaeqiXdqhacaGLOaGaayzkaaGaaGPaVdGaayzc SlaawQa7aiabgkziUoaaBaaaleaacaWGWbaabeaakiaaicdacaGGSa aaaa@5A2D@ 0 < v < τ U . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGimaiabgY da8iaadAhacqGH8aapcqaHepaDdaWgaaWcbaGaamyvaaqabaGccaGG Uaaaaa@3D3A@

Théorème 2. En supposant que les conditions de régularité R.1 à R.7 soient vérifiées, si lim n n 1 / 2 S L = 0 , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaCbeaeaaci GGSbGaaiyAaiaac2gaaSqaaiaad6gacqGHsgIRcqGHEisPaeqaaOGa amOBamaaCaaaleqabaWaaSGbaeaacaaIXaaabaGaaGOmaaaaaaGcda qbdaqaaiaaykW7tuuDJXwAK1uy0HMmaeHbfv3ySLgzG0uy0HgiuD3B aGqbbiab=jj8tnaaBaaaleaacaWGmbaabeaakiaaykW7aiaawMa7ca GLkWoacqGH9aqpcaaIWaGaaiilaaaa@55C3@ alors n 1 / 2 { β ^ ( τ ) β 0 ( τ ) } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBamaaCa aaleqabaWaaSGbaeaacaaIXaaabaGaaGOmaaaaaaGcdaGadaqaaiaa ykW7ceWHYoGbaKaadaqadaqaaiabes8a0bGaayjkaiaawMcaaiabgk HiTiaahk7adaWgaaWcbaGaaGimaaqabaGcdaqadaqaaiabes8a0bGa ayjkaiaawMcaaaGaay5Eaiaaw2haaaaa@476E@ converge faiblement vers un processus gaussien de moyenne zéro pour τ [ v , τ U ] , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiXdqNaey icI48aamWaaeaacaWG2bGaaiilaiaaysW7cqaHepaDdaWgaaWcbaGa amyvaaqabaaakiaawUfacaGLDbaacaGGSaaaaa@41EE@ 0 < v < τ U . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGimaiabgY da8iaadAhacqGH8aapcqaHepaDdaWgaaWcbaGaamyvaaqabaGccaGG Uaaaaa@3D3A@

Pour prouver les théorèmes 1 et 2, nous devons démontrer que max 1 i n | λ θ 0 g ( Z i ; θ 0 ) | = o p ( 1 ) . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaCbeaeaaci GGTbGaaiyyaiaacIhaaSqaaiaaigdacqGHKjYOcaWGPbGaeyizImQa amOBaaqabaGcdaabdaqaaiaaykW7cqaH7oaBdaqhaaWcbaGaeqiUde 3aaSbaaWqaaiaaicdaaeqaaaWcbaqefqvyO9wBHbacfaGaa8hPdaaa kiaadEgadaqadaqaaiaahQfadaWgaaWcbaGaamyAaaqabaGccaGG7a GaaGjbVlaahI7adaWgaaWcbaGaaGimaaqabaaakiaawIcacaGLPaaa caaMc8oacaGLhWUaayjcSdGaeyypa0Jaam4BamaaBaaaleaacaWGWb aabeaakmaabmaabaGaaGymaaGaayjkaiaawMcaaiaac6caaaa@5C9A@ Nous considérons deux types de g ( Z i ; θ ) . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4zamaabm aabaGaaCOwamaaBaaaleaacaWGPbaabeaakiaacUdacaaMe8UaaCiU daGaayjkaiaawMcaaiaac6caaaa@3EB4@ En premier lieu, g ( Z i ; θ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4zamaabm aabaGaaCOwamaaBaaaleaacaWGPbaabeaakiaacUdacaaMe8UaaCiU daGaayjkaiaawMcaaaaa@3E02@ ne contient pas les observations censurées, telle qu’elle est donnée dans l’équation (4.1). Le second type, g ( Z i ; θ ) , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4zamaabm aabaGaaCOwamaaBaaaleaacaWGPbaabeaakiaacUdacaaMe8UaaCiU daGaayjkaiaawMcaaiaacYcaaaa@3EB2@ contient les observations censurées, tel qu’il est donné dans l’équation (4.5).

Dans le cas d’observations non censurées, d’après Owen (1991) et le Lemme 11.2 d’Owen (2001), nous obtenons max 1 i n g ( Z i ; θ 0 ) = o p ( n ) . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaCbeaeaaci GGTbGaaiyyaiaacIhaaSqaaiaaigdacqGHKjYOcaWGPbGaeyizImQa amOBaaqabaGcdaqbdaqaaiaaykW7caWGNbWaaeWaaeaacaWHAbWaaS baaSqaaiaadMgaaeqaaOGaai4oaiaaysW7caWH4oWaaSbaaSqaaiaa icdaaeqaaaGccaGLOaGaayzkaaGaaGPaVdGaayzcSlaawQa7aiabg2 da9iaad+gadaWgaaWcbaGaamiCaaqabaGcdaqadaqaamaakaaabaGa amOBaaWcbeaaaOGaayjkaiaawMcaaiaac6caaaa@54C4@ D’après le Lemme 1 de Tang et Leng (2012), nous obtenons, sous les conditions de régularité R.2, R.3; λ θ 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4UdW2aaS baaSqaaiabeI7aXnaaBaaameaacaaIWaaabeaaaSqabaaaaa@3A7E@ dans l’équation (2.2) satisfait λ θ 0 = O p ( 1 n ) . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaauWaaeaaca aMc8Uaeq4UdW2aaSbaaSqaaiabeI7aXnaaBaaameaacaaIWaaabeaa aSqabaGccaaMc8oacaGLjWUaayPcSdGaeyypa0Jaam4tamaaBaaale aacaWGWbaabeaakmaabmaabaWaaSqaaSqaaiaaigdaaeaadaGcaaqa aiaad6gaaWqabaaaaaGccaGLOaGaayzkaaGaaiOlaaaa@47F5@ Donc,

max 1 i n | λ θ 0 g ( Z i ; θ 0 ) | = O p ( 1 n ) o p ( n ) = o p ( 1 ) . ( 3.4 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaCbeaeaaci GGTbGaaiyyaiaacIhaaSqaaiaaigdacqGHKjYOcaWGPbGaeyizImQa amOBaaqabaGcdaabdaqaaiaaykW7cqaH7oaBdaqhaaWcbaGaeqiUde 3aaSbaaWqaaiaaicdaaeqaaaWcbaqefqvyO9wBHbacfaGaa8hPdaaa kiaadEgadaqadaqaaiaahQfadaWgaaWcbaGaamyAaaqabaGccaGG7a GaaGjbVlaahI7adaWgaaWcbaGaaGimaaqabaaakiaawIcacaGLPaaa caaMc8oacaGLhWUaayjcSdGaeyypa0Jaam4tamaaBaaaleaacaWGWb aabeaakmaabmaabaWaaSaaaeaacaaIXaaabaWaaOaaaeaacaWGUbaa leqaaaaaaOGaayjkaiaawMcaaiaad+gadaWgaaWcbaGaamiCaaqaba GcdaqadaqaamaakaaabaGaamOBaaWcbeaaaOGaayjkaiaawMcaaiab g2da9iaad+gadaWgaaWcbaGaamiCaaqabaGcdaqadaqaaiaaigdaai aawIcacaGLPaaacaGGUaGaaGzbVlaaywW7caaMf8UaaGzbVlaaywW7 caGGOaGaaG4maiaac6cacaaI0aGaaiykaaaa@7317@

Sous la condition R.4, Qin et Jing (2001) ont démontré max 1 i n | λ θ 0 g ( Z i ; θ 0 ) | = o p ( 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaCbeaeaaci GGTbGaaiyyaiaacIhaaSqaaiaaigdacqGHKjYOcaWGPbGaeyizImQa amOBaaqabaGcdaabdaqaaiaaykW7cqaH7oaBdaqhaaWcbaGaeqiUde 3aaSbaaWqaaiaaicdaaeqaaaWcbaqefqvyO9wBHbacfaGaa8hPdaaa kiaadEgadaqadaqaaiaahQfadaWgaaWcbaGaamyAaaqabaGccaGG7a GaaGjbVlaahI7adaWgaaWcbaGaaGimaaqabaaakiaawIcacaGLPaaa caaMc8oacaGLhWUaayjcSdGaeyypa0Jaam4BamaaBaaaleaacaWGWb aabeaakmaabmaabaGaaGymaaGaayjkaiaawMcaaaaa@5BE9@ pour g ( ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4zamaabm aabaGaeyyXICnacaGLOaGaayzkaaaaaa@3AB6@ avec les observations censurées.

Maintenant, d’après Owen (2001), à l’aide du développement en série des poids de Taylor, on peut reformuler les P i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiuamaaBa aaleaacaWGPbaabeaaaaa@37E6@ définis dans l’équation (2.2) sous la forme

P i ( θ 0 ) = 1 n { 1 + λ θ 0 g ( Z i ; θ 0 ) } = 1 n [ 1 λ θ 0 g ( Z i ; θ 0 ) { 1 + o p ( 1 ) } ] = 1 n [ 1 λ θ 0 g ( Z i ; θ 0 ) ] + o p ( 1 n ) ; i = 1 , 2 , , n . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqbaeaabmGaaa qaaiaadcfadaWgaaWcbaGaamyAaaqabaGcdaqadaqaaiaahI7adaWg aaWcbaGaaGimaaqabaaakiaawIcacaGLPaaaaeaacqGH9aqpdaWcaa qaaiaaigdaaeaacaWGUbWaaiWaaeaacaaMc8UaaGymaiabgUcaRiab eU7aSnaaDaaaleaacqaH4oqCdaWgaaadbaGaaGimaaqabaaaleaaru avHH2BTfgaiuaacaWFKoaaaOGaam4zamaabmaabaGaaCOwamaaBaaa leaacaWGPbaabeaakiaacUdacaaMe8UaaCiUdmaaBaaaleaacaaIWa aabeaaaOGaayjkaiaawMcaaaGaay5Eaiaaw2haaaaaaeaaaeaacqGH 9aqpdaWcaaqaaiaaigdaaeaacaWGUbaaamaadmaabaGaaGymaiabgk HiTiabeU7aSnaaDaaaleaacqaH4oqCdaWgaaadbaGaaGimaaqabaaa leaacaWFKoaaaOGaam4zamaabmaabaGaaCOwamaaBaaaleaacaWGPb aabeaakiaacUdacaaMe8UaaCiUdmaaBaaaleaacaaIWaaabeaaaOGa ayjkaiaawMcaamaacmaabaGaaGjcVlaaigdacqGHRaWkcaWGVbWaaS baaSqaaiaadchaaeqaaOWaaeWaaeaacaaIXaaacaGLOaGaayzkaaaa caGL7bGaayzFaaaacaGLBbGaayzxaaaabaaabaGaeyypa0ZaaSaaae aacaaIXaaabaGaamOBaaaadaWadaqaaiaaigdacqGHsislcqaH7oaB daqhaaWcbaGaeqiUde3aaSbaaWqaaiaaicdaaeqaaaWcbaGaa8hPda aakiaadEgadaqadaqaaiaahQfadaWgaaWcbaGaamyAaaqabaGccaGG 7aGaaGjbVlaahI7adaWgaaWcbaGaaGimaaqabaaakiaawIcacaGLPa aaaiaawUfacaGLDbaacqGHRaWkcaWGVbWaaSbaaSqaaiaadchaaeqa aOWaaeWaaeaadaWcaaqaaiaaigdaaeaacaWGUbaaaaGaayjkaiaawM caaiaacUdacaaMf8UaamyAaiabg2da9iaaigdacaGGSaGaaGjbVlaa ikdacaGGSaGaaGjbVlablAciljaacYcacaaMe8UaamOBaiaac6caaa aaaa@9E79@

Nous reformulons maintenant S n ( β , τ ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uamaaBa aaleaacaWGUbaabeaakmaabmaabaGaaCOSdiaacYcacaaMe8UaeqiX dqhacaGLOaGaayzkaaaaaa@3EC1@ sous la forme

S n ( β , τ ) = 1 n i = 1 n [ 1 λ θ 0 g ( Z i ; θ 0 ) ] X i { i ( e X i β ( τ ) ) 0 τ I [ Y i e X i β ( u ) ] d H ( u ) } + o p ( 1 n ) = 1 n i = 1 n X i { i ( e X i β ( τ ) ) 0 τ I [ Y i e X i β ( u ) ] d H ( u ) } 1 n i = 1 n λ θ 0 g ( Z i ; θ 0 ) X i { i ( e X i β ( τ ) ) 0 τ I [ Y i e X i β ( u ) ] d H ( u ) } + o p ( 1 n ) = 1 n i = 1 n X i { i ( e X i β ( τ ) ) 0 τ I [ Y i e X i β ( u ) ] d H ( u ) } 1 n i = 1 n W i { i ( e X i β ( τ ) ) 0 τ I [ Y i e X i β ( u ) ] d H ( u ) } + o p ( 1 n ) . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrViFD0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqbaeaabuGaaa aabaGaam4uamaaBaaaleaacaWGUbaabeaakmaabmaabaGaaCOSdiaa cYcacaaMe8UaeqiXdqhacaGLOaGaayzkaaaabaGaeyypa0ZaaSaaae aacaaIXaaabaGaamOBaaaadaaeWbqaamaadmaabaGaaGymaiabgkHi TiabeU7aSnaaDaaaleaacqaH4oqCdaWgaaadbaGaaGimaaqabaaale aaruavHH2BTfgaiuaacaWFKoaaaOGaam4zamaabmaabaGaaCOwamaa BaaaleaacaWGPbaabeaakiaacUdacaaMe8UaaCiUdmaaBaaaleaaca aIWaaabeaaaOGaayjkaiaawMcaaaGaay5waiaaw2faaiaahIfadaWg aaWcbaGaamyAaaqabaGcdaGadaqaaiaayIW7tuuDJXwAK1uy0HMmae Hbfv3ySLgzG0uy0HgiuD3BaGGbbiab+vrionaaBaaaleaacaWGPbaa beaakmaabmaabaGaamyzamaaCaaaleqabaGaaCiwamaaDaaameaaca WGPbaabaGaa8hPdaaaliaahk7adaqadaqaaiabes8a0bGaayjkaiaa wMcaaaaaaOGaayjkaiaawMcaaiabgkHiTmaapedabaacgaGae0hIWN ealeaacaaIWaaabaGaeqiXdqhaniabgUIiYdGcdaWadaqaaiaadMfa daWgaaWcbaGaamyAaaqabaGccqGHLjYScaWGLbWaaWbaaSqabeaaca WHybWaa0baaWqaaiaadMgaaeaacaWFKoaaaSGaaCOSdmaabmaabaGa amyDaaGaayjkaiaawMcaaaaaaOGaay5waiaaw2faaiaadsgacaWGib WaaeWaaeaacaWG1baacaGLOaGaayzkaaaacaGL7bGaayzFaaGaey4k aSIaam4BamaaBaaaleaacaWGWbaabeaakmaabmaabaWaaSaaaeaaca aIXaaabaGaamOBaaaaaiaawIcacaGLPaaaaSqaaiaadMgacqGH9aqp caaIXaaabaGaamOBaaqdcqGHris5aaGcbaaabaGaeyypa0ZaaSaaae aacaaIXaaabaGaamOBaaaadaaeWbqaaiaahIfadaWgaaWcbaGaamyA aaqabaGcdaGadaqaaiaayIW7cqGFveItdaWgaaWcbaGaamyAaaqaba GcdaqadaqaaiaadwgadaahaaWcbeqaaiaahIfadaqhaaadbaGaamyA aaqaaiaa=r6aaaWccaWHYoWaaeWaaeaacqaHepaDaiaawIcacaGLPa aaaaaakiaawIcacaGLPaaacqGHsisldaWdXaqaaiab9Hi8jbWcbaGa aGimaaqaaiabes8a0bqdcqGHRiI8aOWaamWaaeaacaWGzbWaaSbaaS qaaiaadMgaaeqaaOGaeyyzImRaamyzamaaCaaaleqabaGaaCiwamaa DaaameaacaWGPbaabaGaa8hPdaaaliaahk7adaqadaqaaiaadwhaai aawIcacaGLPaaaaaaakiaawUfacaGLDbaacaWGKbGaamisamaabmaa baGaamyDaaGaayjkaiaawMcaaaGaay5Eaiaaw2haaaWcbaGaamyAai abg2da9iaaigdaaeaacaWGUbaaniabggHiLdaakeaaaeaacaaMe8Ua aGjbVlabgkHiTmaalaaabaGaaGymaaqaaiaad6gaaaWaaabCaeaacq aH7oaBdaqhaaWcbaGaeqiUde3aaSbaaWqaaiaaicdaaeqaaaWcbaGa a8hPdaaakiaadEgadaqadaqaaiaahQfadaWgaaWcbaGaamyAaaqaba GccaGG7aGaaGjbVlaahI7adaWgaaWcbaGaaGimaaqabaaakiaawIca caGLPaaacaWHybWaaSbaaSqaaiaadMgaaeqaaOWaaiWaaeaacaaMi8 Uae4xfH40aaSbaaSqaaiaadMgaaeqaaOWaaeWaaeaacaWGLbWaaWba aSqabeaacaWHybWaa0baaWqaaiaadMgaaeaacaWFKoaaaSGaaCOSdm aabmaabaGaeqiXdqhacaGLOaGaayzkaaaaaaGccaGLOaGaayzkaaGa eyOeI0Yaa8qmaeaacqqFicFsaSqaaiaaicdaaeaacqaHepaDa0Gaey 4kIipakmaadmaabaGaamywamaaBaaaleaacaWGPbaabeaakiabgwMi ZkaadwgadaahaaWcbeqaaiaahIfadaqhaaadbaGaamyAaaqaaiaa=r 6aaaWccaWHYoWaaeWaaeaacaWG1baacaGLOaGaayzkaaaaaaGccaGL BbGaayzxaaGaamizaiaadIeadaqadaqaaiaadwhaaiaawIcacaGLPa aaaiaawUhacaGL9baacqGHRaWkcaWGVbWaaSbaaSqaaiaadchaaeqa aOWaaeWaaeaadaWcaaqaaiaaigdaaeaacaWGUbaaaaGaayjkaiaawM caaaWcbaGaamyAaiabg2da9iaaigdaaeaacaWGUbaaniabggHiLdaa keaaaeaacqGH9aqpdaWcaaqaaiaaigdaaeaacaWGUbaaamaaqahaba GaaCiwamaaBaaaleaacaWGPbaabeaakmaacmaabaGaaGjcVlab+vri onaaBaaaleaacaWGPbaabeaakmaabmaabaGaamyzamaaCaaaleqaba GaaCiwamaaDaaameaacaWGPbaabaGaa8hPdaaaliaahk7adaqadaqa aiabes8a0bGaayjkaiaawMcaaaaaaOGaayjkaiaawMcaaiabgkHiTm aapedabaGae0hIWNealeaacaaIWaaabaGaeqiXdqhaniabgUIiYdGc daWadaqaaiaadMfadaWgaaWcbaGaamyAaaqabaGccqGHLjYScaWGLb WaaWbaaSqabeaacaWHybWaa0baaWqaaiaadMgaaeaacaWFKoaaaSGa aCOSdmaabmaabaGaamyDaaGaayjkaiaawMcaaaaaaOGaay5waiaaw2 faaiaadsgacaWGibWaaeWaaeaacaWG1baacaGLOaGaayzkaaaacaGL 7bGaayzFaaaaleaacaWGPbGaeyypa0JaaGymaaqaaiaad6gaa0Gaey yeIuoaaOqaaaqaaiaaysW7caaMe8UaeyOeI0YaaSaaaeaacaaIXaaa baGaamOBaaaadaaeWbqaaiaahEfadaWgaaWcbaGaamyAaaqabaGcda GadaqaaiaayIW7cqGFveItdaWgaaWcbaGaamyAaaqabaGcdaqadaqa aiaadwgadaahaaWcbeqaaiaahIfadaqhaaadbaGaamyAaaqaaiaa=r 6aaaWccaWHYoWaaeWaaeaacqaHepaDaiaawIcacaGLPaaaaaaakiaa wIcacaGLPaaacqGHsisldaWdXaqaaiab9Hi8jbWcbaGaaGimaaqaai abes8a0bqdcqGHRiI8aOWaamWaaeaacaWGzbWaaSbaaSqaaiaadMga aeqaaOGaeyyzImRaamyzamaaCaaaleqabaGaaCiwamaaDaaameaaca WGPbaabaGaa8hPdaaaliaahk7adaqadaqaaiaadwhaaiaawIcacaGL PaaaaaaakiaawUfacaGLDbaacaWGKbGaamisamaabmaabaGaamyDaa GaayjkaiaawMcaaaGaay5Eaiaaw2haaiabgUcaRiaad+gadaWgaaWc baGaamiCaaqabaGcdaqadaqaamaalaaabaGaaGymaaqaaiaad6gaaa aacaGLOaGaayzkaaGaaiOlaaWcbaGaamyAaiabg2da9iaaigdaaeaa caWGUbaaniabggHiLdaaaaaa@8EAA@

Asymptotiquement, au moyen de l’équation (3.4), nous obtenons W i = o p ( 1 ) ; i = 1 , 2 , , n . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaauWaaeaaca aMc8UaaC4vamaaBaaaleaacaWGPbaabeaakiaaykW7aiaawMa7caGL kWoacqGH9aqpcaWGVbWaaSbaaSqaaiaadchaaeqaaOWaaeWaaeaaca aIXaaacaGLOaGaayzkaaGaai4oaiaaysW7caWGPbGaeyypa0JaaGym aiaacYcacaaMe8UaaGOmaiaacYcacaaMe8UaeSOjGSKaaiilaiaays W7caWGUbGaaiOlaaaa@52D6@ Donc,

S n ( β , τ ) = 1 n i = 1 n X i { i ( e X i β ( τ ) ) 0 τ I [ Y i e X i β ( u ) ] d H ( u ) } + o p ( 1 n ) . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uamaaBa aaleaacaWGUbaabeaakmaabmaabaGaaCOSdiaacYcacaaMe8UaeqiX dqhacaGLOaGaayzkaaGaeyypa0ZaaSaaaeaacaaIXaaabaGaamOBaa aadaaeWbqaaiaahIfadaWgaaWcbaGaamyAaaqabaGcdaGadaqaaiaa yIW7tuuDJXwAK1uy0HMmaeHbfv3ySLgzG0uy0HgiuD3BaGqbbiab=v rionaaBaaaleaacaWGPbaabeaakmaabmaabaGaamyzamaaCaaaleqa baGaaCiwamaaDaaameaacaWGPbaabaqefqvyO9wBHbacgaGaa4hPda aaliaahk7adaqadaqaaiabes8a0bGaayjkaiaawMcaaaaaaOGaayjk aiaawMcaaiabgkHiTmaapedabaacfaGae0hIWNealeaacaaIWaaaba GaeqiXdqhaniabgUIiYdGcdaWadaqaaiaadMfadaWgaaWcbaGaamyA aaqabaGccqGHLjYScaWGLbWaaWbaaSqabeaacaWHybWaa0baaWqaai aadMgaaeaacaGFKoaaaSGaaCOSdmaabmaabaGaamyDaaGaayjkaiaa wMcaaaaaaOGaay5waiaaw2faaiaadsgacaWGibWaaeWaaeaacaWG1b aacaGLOaGaayzkaaaacaGL7bGaayzFaaGaey4kaSIaam4BamaaBaaa leaacaWGWbaabeaakmaabmaabaWaaSaaaeaacaaIXaaabaGaamOBaa aaaiaawIcacaGLPaaacaGGUaaaleaacaWGPbGaeyypa0JaaGymaaqa aiaad6gaa0GaeyyeIuoaaaa@86D1@

Asymptotiquement, cette fonction d’estimation, S n ( β , τ ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uamaaBa aaleaacaWGUbaabeaakmaabmaabaGaaCOSdiaacYcacaaMe8UaeqiX dqhacaGLOaGaayzkaaaaaa@3EC1@ est équivalente à celle formulée dans Peng et Huang (2008). En avançant des arguments semblables à ceux de Peng et Huang (2008), nous achevons la preuve des théorèmes 1 et 2.

Comme l’ont indiqué Peng et Huang (2008), il est difficile de produire une estimation de la variance asymptotique des estimations par régression quantile, car la matrice de covariance du processus limite de n { β ^ ( τ ) β 0 ( τ ) } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOaaaeaaca WGUbaaleqaaOWaaiWaaeaacuaHYoGygaqcamaabmaabaGaeqiXdqha caGLOaGaayzkaaGaeyOeI0IaeqOSdi2aaSbaaSqaaiaaicdaaeqaaO WaaeWaaeaacqaHepaDaiaawIcacaGLPaaaaiaawUhacaGL9baaaaa@450B@ fait intervenir la fonction de densité inconnue f ( y | X ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaabm aabaGaamyEaiaaykW7daabbaqaaiaaykW7caWHybaacaGLhWoaaiaa wIcacaGLPaaaaaa@3EF4@ et f ˜ ( y | X ) . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOzayaaia WaaeWaaeaacaWG5bGaaGPaVpaaeeaabaGaaGPaVlaahIfaaiaawEa7 aaGaayjkaiaawMcaaiaac6caaaa@3FB5@ Plutôt que de recourir à un lissage ou autre approximation numérique, nous proposons une approche bootstrap simple pour produire une estimation des erreurs-types des estimations par régression. Cette approche est utilisée dans notre analyse du rendement examinée à la section suivante.


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