Estimation de quantiles sur petits domaines à l’aide de la régression spline et de la vraisemblance empirique

Section 2. Modèle et hypothèses

Prenons une population finie contenant N= i=0 m   N i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOtaiaai2 dadaaeWaqaaiaabccacaWGobWaaSbaaSqaaiaadMgaaeqaaaqaaiaa dMgacaaI9aGaaGimaaqaaiaad2gaa0GaeyyeIuoaaaa@3F79@ unités d’échantillonnage divisées en m + 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBaiabgU caRiaaigdaaaa@3886@ petits domaines { ( x i j , y i j ) : j = 1, 2, , N i } , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiWaaeaada qadaqaaiaadIhadaWgaaWcbaGaamyAaiaadQgaaeqaaOGaaGilaiaa ysW7caWG5bWaaSbaaSqaaiaadMgacaWGQbaabeaaaOGaayjkaiaawM caaiaaysW7caaI6aGaaGjbVlaaykW7caWGQbGaaGypaiaaigdacaaI SaGaaGjbVlaaikdacaaISaGaaGjbVlablAciljaaiYcacaaMe8Uaam OtamaaBaaaleaacaWGPbaabeaaaOGaay5Eaiaaw2haaiaacYcaaaa@553D@ i = 0, 1 , , m . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAaiaai2 dacaaIWaGaaGilaiaaysW7caaIXaGaaiilaiaaysW7cqWIMaYscaaI SaGaaGjbVlaad2gacaGGUaaaaa@42AA@ Prenons un modèle de régression non paramétrique à erreurs emboîtées avec une covariable :

y i j = m 0 ( x i j ) + v i + ε i j , ( 2.1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEamaaBa aaleaacaWGPbGaamOAaaqabaGccaaI9aGaamyBamaaBaaaleaacaaI WaaabeaakmaabmaabaGaamiEamaaBaaaleaacaWGPbGaamOAaaqaba aakiaawIcacaGLPaaacqGHRaWkcaWG2bWaaSbaaSqaaiaadMgaaeqa aOGaey4kaSIaeqyTdu2aaSbaaSqaaiaadMgacaWGQbaabeaakiaaiY cacaaMf8UaaGzbVlaaywW7caaMf8UaaGzbVlaacIcacaaIYaGaaiOl aiaaigdacaGGPaaaaa@53E4@

x i j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBa aaleaacaWGPbGaamOAaaqabaaaaa@38FD@ est une variable auxiliaire, v i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODamaaBa aaleaacaWGPbaabeaaaaa@380C@ désigne an effet aléatoire propre au domaine et ε i j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyTdu2aaS baaSqaaiaadMgacaWGQbaabeaaaaa@39A7@ représente des erreurs aléatoires. La fonction de régression m 0 ( ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBamaaBa aaleaacaaIWaaabeaakmaabmaabaGaeyyXICnacaGLOaGaayzkaaaa aa@3BAC@ n’est pas précisée, mais nous pouvons calculer assez facilement une approximation à l’aide d’une fonction spline

m 0 ( x ; β , γ ) = β 0 + β 1 x + + β p x p + k = 1 K γ k ( x κ k ) + p . ( 2.2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBamaaBa aaleaacaaIWaaabeaakmaabmaabaGaamiEaiaaiUdacaaMe8UaaCOS diaaiYcacaaMe8UaaC4SdaGaayjkaiaawMcaaiaai2dacqaHYoGyda WgaaWcbaGaaGimaaqabaGccqGHRaWkcqaHYoGydaWgaaWcbaGaaGym aaqabaGccaWG4bGaey4kaSIaeSOjGSKaey4kaSIaeqOSdi2aaSbaaS qaaiaadchaaeqaaOGaamiEamaaCaaaleqabaGaamiCaaaakiabgUca RmaaqahabaGaeq4SdC2aaSbaaSqaaiaadUgaaeqaaOWaaeWaaeaaca WG4bGaeyOeI0IaeqOUdS2aaSbaaSqaaiaadUgaaeqaaaGccaGLOaGa ayzkaaWaa0baaSqaaiabgUcaRaqaaiaadchaaaaabaGaam4Aaiaai2 dacaaIXaaabaGaam4saaqdcqGHris5aOGaaGOlaiaaywW7caaMf8Ua aGzbVlaaywW7caaMf8UaaiikaiaaikdacaGGUaGaaGOmaiaacMcaaa a@6E96@

Ici, p MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCaaaa@36EC@ est le degré de l’estimation spline, x + p = x p MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaDa aaleaacqGHRaWkaeaacaWGWbaaaOGaaGypaiaadIhadaahaaWcbeqa aiaadchaaaaaaa@3BE8@ lorsque x > 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiaai6 dacaaIWaaaaa@3876@ et 0 autrement, κ k , k = 1, , K MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOUdS2aaS baaSqaaiaadUgaaeqaaOGaaGilaiaaysW7caWGRbGaaGypaiaaigda caaISaGaaGjbVlablAciljaaiYcacaaMe8Uaam4saaaa@43FC@ constituent un ensemble de constantes fixes appelées nœuds, β = ( β 0 , , β p ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCOSdiaai2 dadaqadaqaaiabek7aInaaBaaaleaacaaIWaaabeaakiaaiYcacaaM e8UaeSOjGSKaaGilaiaaysW7cqaHYoGydaWgaaWcbaGaamiCaaqaba aakiaawIcacaGLPaaadaahaaWcbeqaaOGamai2gkdiIcaaaaa@47A1@ est un vecteur de coefficient de la portion paramétrique du modèle, et γ = ( γ 1 , , γ K ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaC4Sdiaai2 dadaqadaqaaiabeo7aNnaaBaaaleaacaaIXaaabeaakiaaiYcacaaM e8UaeSOjGSKaaGilaiaaysW7cqaHZoWzdaWgaaWcbaGaam4saaqaba aakiaawIcacaGLPaaadaahaaWcbeqaaOGamai2gkdiIcaaaaa@478A@ est le vecteur des coefficients splines, K MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4saaaa@36C7@ est le nombre de nœuds splines. Si l’emplacement des noeuds couvre toute la plage de x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaaaa@36F4@ et K MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4saaaa@36C7@ est suffisamment important, la classe de P-spline (2.2) peut servir à estimer une fonction lisse m 0 ( ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBamaaBa aaleaacaaIWaaabeaakmaabmaabaGaeyyXICnacaGLOaGaayzkaaaa aa@3BAC@ avec un niveau d’exactitude élevé, même pour un petit p MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCaaaa@36EC@ (Boor, 2001). Ruppert, Wand et Carroll (2003) ont recommandé d’utiliser le nombre de nœuds splines K MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4saaaa@36C7@ comme le minimum de 40 et le nombre de x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaaaa@36F4@ uniques divisé par 4.

Nous supposons que nous obtenons un échantillon aléatoire de la population à l’aide d’un plan de sondage non informatif, de telle sorte que (2.1) demeure valide pour les unités échantillonnées. Notre tâche immédiate consiste à ajuster ce modèle à partir des données échantillonnées et de suivre l’approche d’Opsomer et coll. (2008). Pour simplifier la présentation, nous introduisons d’abord une notation matricielle. Supposons que n i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBamaaBa aaleaacaWGPbaabeaaaaa@3804@ est le nombre d’unités échantillonnées dans le petit domaine i . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAaiaac6 caaaa@3797@ Les valeurs de réponses des i e MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAamaaCa aaleqabaGaaeyzaaaaaaa@37FA@ domaines sont désignées comme y i = ( y i 1 , y i 2 , , y i n i ) . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCyEamaaBa aaleaacaWGPbaabeaakiaai2dadaqadaqaaiaadMhadaWgaaWcbaGa amyAaiaaigdaaeqaaOGaaGilaiaaysW7caWG5bWaaSbaaSqaaiaadM gacaaIYaaabeaakiaaiYcacaaMe8UaeSOjGSKaaGilaiaaysW7caWG 5bWaaSbaaSqaaiaadMgacaWGUbWaaSbaaWqaaiaadMgaaeqaaaWcbe aaaOGaayjkaiaawMcaamaaCaaaleqabaGccWaGyBOmGikaaiaac6ca aaa@5017@ Nous les regroupons ensuite pour créer le vecteur de réponse dont la longueur est n : MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaiaays W7caGG6aaaaa@3935@ Y n = ( y 0 , y 1 , , y m ) . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCywamaaDa aaleaacaWGUbaabaqcLbwacWaGyBOmGikaaOGaaGypamaabmaabaGa aCyEamaaDaaaleaacaaIWaaabaqcLbwacWaGyBOmGikaaOGaaGilai aaysW7caWH5bWaa0baaSqaaiaaigdaaeaajugybiadaITHYaIOaaGc caaISaGaaGjbVlablAciljaaiYcacaaMe8UaaCyEamaaDaaaleaaca WGTbaabaqcLbwacWaGyBOmGikaaaGccaGLOaGaayzkaaGaaiOlaaaa @57BE@ De même, nous définissons ϵ i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWefv3ySLgznf gDOfdaryqr1ngBPrginfgDObYtUvgaiuqacqWF1pG8daWgaaWcbaGa amyAaaqabaaaaa@430A@ et ϵ n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWefv3ySLgznf gDOfdaryqr1ngBPrginfgDObYtUvgaiuqacqWF1pG8daWgaaWcbaGa amOBaaqabaaaaa@4310@ pour le terme d’erreur. Nous utilisons v = ( v 0 , , v m ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCODaiaai2 dadaqadaqaaiaadAhadaWgaaWcbaGaaGimaaqabaGccaaISaGaaGjb VlablAciljaaiYcacaaMe8UaamODamaaBaaaleaacaWGTbaabeaaaO GaayjkaiaawMcaamaaCaaaleqabaGccWaGyBOmGikaaaaa@4613@ pour les effets aléatoires propres à un domaine et créons une matrice D MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCiraaaa@36C4@ de sorte que

D v = ( v 0 1 n 0 , v 1 1 n 1 , , v m 1 n m ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCiraiaahA hacaaI9aGaaGikaiaadAhadaWgaaWcbaGaaGimaaqabaqefCuzVj3z Pfgaiuaakiaa=fdadaqhaaWcbaGaamOBamaaBaaameaacaaIWaaabe aaaSqaaKqzGfGamai2gkdiIcaakiaaiYcacaaMe8UaamODamaaBaaa leaacaaIXaaabeaakiaa=fdadaqhaaWcbaGaamOBamaaBaaameaaca aIXaaabeaaaSqaaKqzGfGamai2gkdiIcaakiaaiYcacaaMe8UaeSOj GSKaaGilaiaaysW7caWG2bWaaSbaaSqaaiaad2gaaeqaaOGaa8xmam aaDaaaleaacaWGUbWaaSbaaWqaaiaad2gaaeqaaaWcbaqcLbwacWaG yBOmGikaaOGaaGykaaaa@5E64@

1 k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqefCuzVj3zPf gaiuaacaWFXaWaaSbaaSqaaiaadUgaaeqaaaaa@3AA6@ est un vecteur de longueur k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4Aaaaa@36E7@ de 1. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiaab6 caaaa@3762@ Nous construisons ensuite des matrices X n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCiwamaaBa aaleaacaWGUbaabeaaaaa@37F7@ et Z n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCOwamaaBa aaleaacaWGUbaabeaaaaa@37F9@ de telle sorte que leurs rangées soient composées de

x i j = ( 1, x i j , , x i j p ) , z i j = ( ( x i j κ 1 ) + p , , ( x i j κ K ) + p ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCiEamaaDa aaleaacaWGPbGaamOAaaqaaKqzGfGamai2gkdiIcaakiaai2dadaqa daqaaiaaigdacaaISaGaaGjbVlaadIhadaWgaaWcbaGaamyAaiaadQ gaaeqaaOGaaGilaiaaysW7cqWIMaYscaaISaGaaGjbVlaadIhadaqh aaWcbaGaamyAaiaadQgaaeaacaWGWbaaaaGccaGLOaGaayzkaaGaaG ilaiaaywW7caWH6bWaa0baaSqaaiaadMgacaWGQbaabaqcLbwacWaG yBOmGikaaOGaaGypamaabmaabaWaaeWaaeaacaWG4bWaaSbaaSqaai aadMgacaWGQbaabeaakiabgkHiTiabeQ7aRnaaBaaaleaacaaIXaaa beaaaOGaayjkaiaawMcaamaaDaaaleaacqGHRaWkaeaacaWGWbaaaO GaaGilaiaaysW7cqWIMaYscaaISaGaaGjbVpaabmaabaGaamiEamaa BaaaleaacaWGPbGaamOAaaqabaGccqGHsislcqaH6oWAdaWgaaWcba Gaam4saaqabaaakiaawIcacaGLPaaadaqhaaWcbaGaey4kaScabaGa amiCaaaaaOGaayjkaiaawMcaaaaa@744B@

dans l’ordre approprié. Avec ces matrices et ces vecteurs, les données de l’échantillon sous le modèle (2.1) sont reliées par

Y n = X n β + Z n γ + D v + ϵ n . ( 2.3 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCywamaaBa aaleaacaWGUbaabeaakiaai2dacaWHybWaaSbaaSqaaiaad6gaaeqa aOGaaCOSdiabgUcaRiaahQfadaWgaaWcbaGaamOBaaqabaGccaWHZo Gaey4kaSIaaCiraiaahAhacqGHRaWktuuDJXwAK1uy0HwmaeHbfv3y SLgzG0uy0Hgip5wzaGqbbiab=v=aYpaaBaaaleaacaWGUbaabeaaki aai6cacaaMf8UaaGzbVlaaywW7caaMf8UaaGzbVlaacIcacaaIYaGa aiOlaiaaiodacaGGPaaaaa@5CF2@

Opsomer et coll. (2008) ont ajusté ce modèle selon l’hypothèse voulant que les composantes de γ , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaC4SdiaacY caaaa@37E6@ de v MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCODaaaa@36F6@ et de ϵ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWefv3ySLgznf gDOfdaryqr1ngBPrginfgDObYtUvgaiuqacqWF1pG8aaa@41F1@ soient toutes indépendantes et normalement distribuées de manière identique avec les variances σ γ 2 , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aa0 baaSqaaiabeo7aNbqaaiaaikdaaaGccaGGSaaaaa@3B04@ σ v 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aa0 baaSqaaiaadAhaaeaacaaIYaaaaaaa@399E@ et σ ϵ 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aa0 baaSqaamrr1ngBPrwtHrhAXaqeguuDJXwAKbstHrhAG8KBLbacfaGa e8x9dipabaGaaGOmaaaaaaa@449C@ respectivement. Nous obtenons les solutions de l’ajustement comme suit

V ^ = Z n Σ γ Z n + D Σ ^ v D + Σ ^ ε , v ^ = Σ ^ v D V ^ 1 ( Y n X n β ^ ) , β ^ = ( X n V ^ 1 X n ) 1 ( X n V ^ 1 Y n ) , γ ^ = Σ ^ γ Z n V ^ 1 ( Y n X n β ^ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrVipC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqbaeaabqGaaa aabaGabCOvayaajaaabaGaaGypaiaahQfadaWgaaWcbaGaamOBaaqa baGccaWHJoWaaSbaaSqaaiabeo7aNbqabaGccaWHAbWaa0baaSqaai aad6gaaeaajugybiadaITHYaIOaaGccqGHRaWkcaWHebGabC4Odyaa jaWaaSbaaSqaaiaadAhaaeqaaOGabCirayaafaGaey4kaSIabC4Ody aajaWaaSbaaSqaaiabew7aLbqabaGccaaISaaabaGabCODayaajaaa baGaaGypaiqaho6agaqcamaaBaaaleaacaWG2baabeaakiqahseaga qbaiqahAfagaqcamaaCaaaleqabaGaeyOeI0IaaGymaaaakmaabmaa baGaaCywamaaBaaaleaacaWGUbaabeaakiabgkHiTiaahIfadaWgaa WcbaGaamOBaaqabaGcceWHYoGbaKaaaiaawIcacaGLPaaacaaISaaa baGabCOSdyaajaaabaGaaGypamaabmaabaGaaCiwamaaDaaaleaaca WGUbaabaqcLbwacWaGyBOmGikaaOGabCOvayaajaWaaWbaaSqabeaa cqGHsislcaaIXaaaaOGaaCiwamaaBaaaleaacaWGUbaabeaaaOGaay jkaiaawMcaamaaCaaaleqabaGaeyOeI0IaaGymaaaakmaabmaabaGa aCiwamaaDaaaleaacaWGUbaabaqcLbwacWaGyBOmGikaaOGabCOvay aajaWaaWbaaSqabeaacqGHsislcaaIXaaaaOGaaCywamaaBaaaleaa caWGUbaabeaaaOGaayjkaiaawMcaaiaaiYcaaeaaceWHZoGbaKaaae aacaaI9aGabC4OdyaajaWaaSbaaSqaaiabeo7aNbqabaGccaWHAbWa a0baaSqaaiaad6gaaeaajugybiadaITHYaIOaaGcceWHwbGbaKaada ahaaWcbeqaaiabgkHiTiaaigdaaaGcdaqadaqaaiaahMfadaWgaaWc baGaamOBaaqabaGccqGHsislcaWHybWaaSbaaSqaaiaad6gaaeqaaO GabCOSdyaajaaacaGLOaGaayzkaaaaaaaa@8F3D@

Σ ^ γ , Σ ^ v , Σ ^ ε MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabC4Odyaaja WaaSbaaSqaaiabeo7aNbqabaGccaaISaGaaGjbVlqaho6agaqcamaa BaaaleaacaWG2baabeaakiaaiYcacaaMe8UabC4OdyaajaWaaSbaaS qaaiabew7aLbqabaaaaa@431B@ sont des estimations du maximum de vraisemblance restreinte (REML) pour les matrices de covariance de γ , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaC4SdiaacY caaaa@37E6@ v MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCODaaaa@36F6@ et ϵ , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWefv3ySLgznf gDOfdaryqr1ngBPrginfgDObYtUvgaiuqacqWF1pG8caGGSaaaaa@42A1@ et V ^ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabCOvayaaja aaaa@36E6@ est l’estimation de V var ( Y n ) . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCOvaiabgg Mi6kGacAhacaGGHbGaaiOCamaabmaabaGaaCywamaaBaaaleaacaWG UbaabeaaaOGaayjkaiaawMcaaiaac6caaaa@3FBB@

Opsomer et coll. (2008) ont ensuite établi la meilleure prédiction linéaire sans biais empirique de la moyenne de petit domaine :

Y ¯ ^ i = β ^ 0 + β ^ 1 X ¯ i + + β ^ p X ¯ i p + z ¯ i γ ^ + v ^ i , ( 2.4 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmywayaary aajaWaaSbaaSqaaiaadMgaaeqaaOGaaGypaiqbek7aIzaajaWaaSba aSqaaiaaicdaaeqaaOGaey4kaSIafqOSdiMbaKaadaWgaaWcbaGaaG ymaaqabaGcceWGybGbaebadaWgaaWcbaGaamyAaaqabaGccqGHRaWk cqWIMaYscqGHRaWkcuaHYoGygaqcamaaBaaaleaacaWGWbaabeaaki qadIfagaqeamaaDaaaleaacaWGPbaabaGaamiCaaaakiabgUcaRiqa hQhagaqeamaaBaaaleaacaWGPbaabeaakiqaho7agaqcaiabgUcaRi qadAhagaqcamaaBaaaleaacaWGPbaabeaakiaaiYcacaaMf8UaaGzb VlaaywW7caaMf8UaaGzbVlaacIcacaaIYaGaaiOlaiaaisdacaGGPa aaaa@5D77@

X ¯ i , , X ¯ i p MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmiwayaara WaaSbaaSqaaiaadMgaaeqaaOGaaGilaiaaysW7cqWIMaYscaaISaGa aGjbVlqadIfagaqeamaaDaaaleaacaWGPbaabaGaamiCaaaaaaa@40BD@ représentent les moyennes de puissances des unités de population x i j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBa aaleaacaWGPbGaamOAaaqabaaaaa@38FD@ dans le domaine i , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAaiaacY caaaa@3795@ c’est-à-dire, X ¯ i s = N i 1 j = 1 N i x i j s MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmiwayaara Waa0baaSqaaiaadMgaaeaacaWGZbaaaOGaaGypaiaad6eadaqhaaWc baGaamyAaaqaaiabgkHiTiaaigdaaaGcdaaeWaqaaiaadIhadaqhaa WcbaGaamyAaiaadQgaaeaacaWGZbaaaaqaaiaadQgacaaI9aGaaGym aaqaaiaad6eadaWgaaadbaGaamyAaaqabaaaniabggHiLdaaaa@47C5@ pour s = 1, , p , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4Caiaai2 dacaaIXaGaaGilaiaaysW7cqWIMaYscaaISaGaaGjbVlaadchacaGG Saaaaa@3FBE@ et z ¯ i γ ^ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabCOEayaara WaaSbaaSqaaiaadMgaaeqaaOGabC4Sdyaajaaaaa@3985@ désigne les vraies moyennes des fonctions de base de spline pour le petit domaine i . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAaiaac6 caaaa@3797@ De toute évidence, il est facile d’élargir la discussion qui précède à des modèles additifs non paramétriques comportant deux covariables ou plus (Lin et Zhang (1999), Ruppert et coll. (2003) et Wood (2006)).

Dans le présent document, nous suivons les travaux d’Opsomer et coll. (2008) pour obtenir toutes les valeurs ajustées. Pour l’estimation de quantiles sur petits domaines, nous supprimons l’hypothèse de la normalité pour ϵ i j . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWefv3ySLgznf gDOfdaryqr1ngBPrginfgDObYtUvgaiuaacqWF1pG8daWgaaWcbaGa amyAaiaadQgaaeqaaOGaaiOlaaaa@44B5@ Nous supposons plutôt que leur distribution G i ( u ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4ramaaBa aaleaacaWGPbaabeaakmaabmaabaGaamyDaaGaayjkaiaawMcaaaaa @3A6A@ satisfait un MRD, de telle sorte que pour i = 1, , m , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAaiaai2 dacaaIXaGaaGilaiaaysW7cqWIMaYscaaISaGaaGjbVlaad2gacaGG Saaaaa@3FB1@

log { d G i ( u ) / d G 0 ( u ) } = θ i q ( u ) , ( 2.5 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciiBaiaac+ gacaGGNbWaaiWaaeaadaWcgaqaaiaadsgacaWGhbWaaSbaaSqaaiaa dMgaaeqaaOWaaeWaaeaacaWG1baacaGLOaGaayzkaaaabaGaamizai aadEeadaWgaaWcbaGaaGimaaqabaGcdaqadaqaaiaadwhaaiaawIca caGLPaaaaaaacaGL7bGaayzFaaGaaGypaiabeI7aXnaaDaaaleaaca WGPbaabaqcLbwacWaGyBOmGikaaOGaaCyCamaabmaabaGaamyDaaGa ayjkaiaawMcaaiaaiYcacaaMf8UaaGzbVlaaywW7caaMf8UaaGzbVl aacIcacaaIYaGaaiOlaiaaiwdacaGGPaaaaa@5C61@

avec une fonction de base prédéterminée q ( u ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCyCamaabm aabaGaamyDaaGaayjkaiaawMcaaaaa@3974@ et un paramètre de basculement propre au domaine θ i . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiUde3aaS baaSqaaiaadMgaaeqaaOGaaiOlaaaa@3983@ Nous pouvons inclure i = 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAaiaai2 dacaaIWaaaaa@3866@ dans l’équation ci-haut en posant θ 0 = 0. MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiUde3aaS baaSqaaiaaicdaaeqaaOGaaGypaiaaicdacaGGUaaaaa@3AD0@ Il faut que le premier élément de q ( u ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCyCamaabm aabaGaamyDaaGaayjkaiaawMcaaaaa@3974@ soit un pour que le premier élément de θ i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiUde3aaS baaSqaaiaadMgaaeqaaaaa@38C7@ soit un paramètre de normalisation. Le MRD comprend des familles de distributions normale, gamma et bien d’autres familles comme des cas spéciaux. Des discussions sur le MRD se trouvent dans Anderson (1979), Qin et Zhang (1997), Kezioua et Leoni-Aubina (2008) et Chen et Liu (2013).

Les équations (2.1), (2.2) et (2.5) forment ensemble la plateforme du présent document pour l’estimation de quantiles sur petits domaines. Nos travaux sont différents de ceux d’Opsomer et coll. (2008) en ce sens que nous nous concentrons sur l’estimation de quantiles sur petits domaines sans énoncer d’hypothèse de normalité pour G i ( ) . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4ramaaBa aaleaacaWGPbaabeaakmaabmaabaGaeyyXICnacaGLOaGaayzkaaGa aiOlaaaa@3C6C@ Parallèlement, le présent document est différent de Chen et Liu (2018) parce qu’il formule des postulats pour une relation de régression non paramétrique entre y i j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEamaaBa aaleaacaWGPbGaamOAaaqabaaaaa@38FE@ et x i j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBa aaleaacaWGPbGaamOAaaqabaaaaa@38FD@ au lieu d’une relation linéaire.


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