Régression quantile censurée pondérée

Section 2. Estimation des poids au moyen de la vraisemblance empirique

Nous élaborons une méthode qui permet de convertir l’information auxiliaire en probabilités axées sur des données fondées sur la VE, qui sont ensuite utilisées comme poids dans la régression quantile censurée pondérée. Qin et Lawless (1994) ont élaboré l’approche de la VE d’après un ensemble d’équations d’estimation. Supposons que { Z i } i = 1 n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiWaaeaaca WHAbWaaSbaaSqaaiaadMgaaeqaaaGccaGL7bGaayzFaaWaa0baaSqa aiaadMgacqGH9aqpcaaIXaaabaGaamOBaaaaaaa@3DFE@ représente les données observées et que l’information auxiliaire disponible est représentée par la fonction d’estimation g ( Z i ; θ ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4zamaabm aabaGaaCOwamaaBaaaleaacaWGPbaabeaakiaacUdacaaMe8UaaCiU daGaayjkaiaawMcaaaaa@3E03@ ayant le paramètre, θ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCiUdaaa@373B@ lequel est connu. La vraisemblance empirique maximale est alors donnée par

L VE ( θ ) = sup { i = 1 n P i : P i 0 , i = 1 n P i = 1 , i = 1 n P i g ( Z i ; θ ) = 0 } , ( 2.1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamitamaaBa aaleaacaqGwbGaaeyraaqabaGcdaqadaqaaiaahI7aaiaawIcacaGL PaaacqGH9aqpciGGZbGaaiyDaiaacchadaGadaqaamaarahabaGaam iuamaaBaaaleaacaWGPbaabeaakiaaysW7caGG6aGaaGjbVlaadcfa daWgaaWcbaGaamyAaaqabaGccqGHLjYScaaIWaGaaiilaiaaysW7aS qaaiaadMgacqGH9aqpcaaIXaaabaGaamOBaaqdcqGHpis1aOWaaabC aeaacaWGqbWaaSbaaSqaaiaadMgaaeqaaOGaeyypa0JaaGymaiaacY caaSqaaiaadMgacqGH9aqpcaaIXaaabaGaamOBaaqdcqGHris5aOGa aGjbVpaaqahabaGaamiuamaaBaaaleaacaWGPbaabeaakiaadEgada qadaqaaiaahQfadaWgaaWcbaGaamyAaaqabaGccaGG7aGaaGjbVlaa hI7aaiaawIcacaGLPaaacqGH9aqpcaaIWaaaleaacaWGPbGaeyypa0 JaaGymaaqaaiaad6gaa0GaeyyeIuoaaOGaay5Eaiaaw2haaiaacYca caaMf8UaaGzbVlaaywW7caaMf8UaaGzbVlaacIcacaaIYaGaaiOlai aaigdacaGGPaaaaa@7D7C@

P i = Pr ( Z i = z i ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiuamaaBa aaleaacaWGPbaabeaakiabg2da9iGaccfacaGGYbWaaeWaaeaacaWG AbWaaSbaaSqaaiaadMgaaeqaaOGaeyypa0JaamOEamaaBaaaleaaca WGPbaabeaaaOGaayjkaiaawMcaaaaa@4177@ et θ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCiUdaaa@373B@ est le paramètre de l’information auxiliaire qui peut être présumé connu. Le paramètre, θ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCiUdaaa@373B@ pourrait représenter toute information paramétrique présentée dans les études antérieures qui exerce une influence sur le paramètre du modèle, β ( τ ) . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCOSdmaabm aabaGaeqiXdqhacaGLOaGaayzkaaGaaiOlaaaa@3B35@ Pour une fonction g ( Z i ; θ ) , θ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4zamaabm aabaGaaCOwamaaBaaaleaacaWGPbaabeaakiaacUdacaaMe8UaaCiU daGaayjkaiaawMcaaiaacYcacaaMe8UaaCiUdaaa@4184@ doit remplir la condition E { g ( Z i ; θ ) } = 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyramaacm aabaGaam4zamaabmaabaGaaCOwamaaBaaaleaacaWGPbaabeaakiaa cUdacaaMe8UaaCiUdaGaayjkaiaawMcaaaGaay5Eaiaaw2haaiabg2 da9iaaicdaaaa@42BE@ pour éviter l’absence de solutions découlant de l’enveloppe convexe. Il s’agit du scénario dans lequel zéro n’est pas un point intérieur de l’enveloppe convexe de la fonction g ( Z i ; θ ) , i = 1 , 2 , , n , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4zamaabm aabaGaaCOwamaaBaaaleaacaWGPbaabeaakiaacUdacaaMe8UaaCiU daGaayjkaiaawMcaaiaacYcacaaMe8UaamyAaiabg2da9iaaigdaca GGSaGaaGjbVlaaikdacaGGSaGaaGjbVlablAciljaacYcacaaMe8Ua amOBaiaacYcaaaa@4D27@ qui ne donnera pas de valeurs positives pour les P i . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiuamaaBa aaleaacaWGPbaabeaaieaakiaa=5caaaa@38A8@ Pour une valeur donnée de θ = θ 0 , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCiUdiabg2 da9iaahI7adaWgaaWcbaGaaGimaaqabaGccaGGSaaaaa@3B25@ à l’aide de la méthode des multiplicateurs de Lagrange, L VE ( θ 0 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamitamaaBa aaleaacaqGwbGaaeyraaqabaGcdaqadaqaaiaahI7adaWgaaWcbaGa aGimaaqabaaakiaawIcacaGLPaaaaaa@3C5C@ atteint sa valeur maximale à

P i ( θ 0 ) = 1 n { 1 + λ θ 0 g ( Z i ; θ 0 ) } , i = 1 , 2 , , n . ( 2.2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiuamaaBa aaleaacaWGPbaabeaakmaabmaabaGaaCiUdmaaBaaaleaacaaIWaaa beaaaOGaayjkaiaawMcaaiabg2da9maalaaabaGaaGymaaqaaiaad6 gadaGadaqaaiaaigdacqGHRaWkcqaH7oaBdaqhaaWcbaGaeqiUde3a aSbaaWqaaiaaicdaaeqaaaWcbaqefqvyO9wBHbacfaGaa8hPdaaaki aadEgadaqadaqaaiaahQfadaWgaaWcbaGaamyAaaqabaGccaGG7aGa aGjbVlaahI7adaWgaaWcbaGaaGimaaqabaaakiaawIcacaGLPaaaai aawUhacaGL9baaaaGaaiilaiaaywW7caWGPbGaeyypa0JaaGymaiaa cYcacaaMe8UaaGOmaiaacYcacaaMe8UaeSOjGSKaaiilaiaaysW7ca WGUbGaaiOlaiaaywW7caaMf8UaaGzbVlaaywW7caaMf8Uaaiikaiaa ikdacaGGUaGaaGOmaiaacMcaaaa@6DE2@

Le multiplicateur de Lagrange, λ θ 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4UdW2aaS baaSqaaiabeI7aXnaaBaaameaacaaIWaaabeaaaSqabaaaaa@3A7F@ est la solution à l’équation

i = 1 n g ( Z i ; θ 0 ) n { 1 + λ θ 0 g ( Z i ; θ 0 ) } = 0. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaabCaeaada WcaaqaaiaadEgadaqadaqaaiaahQfadaWgaaWcbaGaamyAaaqabaGc caGG7aGaaGjbVlaahI7adaWgaaWcbaGaaGimaaqabaaakiaawIcaca GLPaaaaeaacaWGUbWaaiWaaeaacaaIXaGaey4kaSIaeq4UdW2aa0ba aSqaaiabeI7aXnaaBaaameaacaaIWaaabeaaaSqaaerbufgAV1wyaG qbaiaa=r6aaaGccaWGNbWaaeWaaeaacaWHAbWaaSbaaSqaaiaadMga aeqaaOGaai4oaiaaysW7caWH4oWaaSbaaSqaaiaaicdaaeqaaaGcca GLOaGaayzkaaaacaGL7bGaayzFaaaaaaWcbaGaamyAaiabg2da9iaa igdaaeaacaWGUbaaniabggHiLdGccqGH9aqpcaaIWaGaaiOlaaaa@5D57@

Les P i ( ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiuamaaBa aaleaacaWGPbaabeaakmaabmaabaGaeyyXICnacaGLOaGaayzkaaaa aa@3BC3@ estimés sont utilisés comme poids ( ω i ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaacq aHjpWDdaWgaaWcbaGaamyAaaqabaaakiaawIcacaGLPaaaaaa@3A71@ dans l’équation (1.2) pour la régression quantile censurée pondérée. Dans certains cas, θ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCiUdaaa@373B@ pourrait ne pas être disponible et, dans une telle situation, on peut utiliser une estimation de θ , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCiUdiaacY caaaa@37EB@ disons θ ^ A , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabCiUdyaaja WaaSbaaSqaaiaadgeaaeqaaOGaaiilaaaa@38F7@ obtenue à partir d’études antérieures. Chen et Qin (1993) ont démontré que, pour un échantillon aléatoire, Y i , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamywamaaBa aaleaacaWGPbaabeaakiaacYcaaaa@38A9@ et les P i ( ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiuamaaBa aaleaacaWGPbaabeaakmaabmaabaGaeyyXICnacaGLOaGaayzkaaaa aa@3BC3@ sont estimés à l’aide de l’équation (2.2), F ˜ n ( y ) = i = 1 n P i I ( Y i y ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaia WaaSbaaSqaaiaad6gaaeqaaOWaaeWaaeaacaWG5baacaGLOaGaayzk aaGaeyypa0ZaaabmaeaacaWGqbWaaSbaaSqaaiaadMgaaeqaamrr1n gBPrwtHrhAYaqeguuDJXwAKbstHrhAGq1DVbacfaGccqWFicFsdaqa daqaaiaadMfadaWgaaWcbaGaamyAaaqabaGccqGHKjYOcaWG5baaca GLOaGaayzkaaaaleaacaWGPbGaeyypa0JaaGymaaqaaiaad6gaa0Ga eyyeIuoaaaa@554B@ présente une variance plus faible que la fonction de distribution empirique, F ^ n ( y ) = 1 n i = 1 n I ( Y i y ) . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaja WaaSbaaSqaaiaad6gaaeqaaOWaaeWaaeaacaWG5baacaGLOaGaayzk aaGaeyypa0ZaaSqaaSqaaiaaigdaaeaacaWGUbaaaOWaaabmaeaatu uDJXwAK1uy0HMmaeHbfv3ySLgzG0uy0HgiuD3BaGqbaiab=Hi8jnaa bmaabaGaamywamaaBaaaleaacaWGPbaabeaakiabgsMiJkaadMhaai aawIcacaGLPaaaaSqaaiaadMgacqGH9aqpcaaIXaaabaGaamOBaaqd cqGHris5aOGaaiOlaaaa@55E3@ Par conséquent, à l’aide d’une représentation de Bahadur (Bahadur, 1966), pour un certain τ ( 0 < τ < 1 ) , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiXdq3aae WaaeaacaaIWaGaeyipaWJaeqiXdqNaeyipaWJaaGymaaGaayjkaiaa wMcaaiaacYcaaaa@3F37@ l’estimation des quantiles, F ˜ n 1 ( τ ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaia Waa0baaSqaaiaad6gaaeaacqGHsislcaaIXaaaaOWaaeWaaeaacqaH epaDaiaawIcacaGLPaaaaaa@3CF1@ présente une variance plus faible que F ^ n 1 ( τ ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaja Waa0baaSqaaiaad6gaaeaacqGHsislcaaIXaaaaOWaaeWaaeaacqaH epaDaiaawIcacaGLPaaaaaa@3CF2@ (voir Kuk et Mak, 1989; Rao et coll., 1990). Ainsi, la méthode que nous proposons devrait être plus efficace que la simple régression quantile censurée.


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