Élaboration d’un système d’estimation sur petits domaines à Statistique Canada

Section 3. Modèle au niveau du domaine

L’estimateur de petits domaines au niveau du domaine est apparu pour la première fois dans le texte fondateur de Fay et Herriot (1979). Pour donner suite à ce document, supposons que le paramètre d’intérêt est θ i ; MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiUde3aaS baaSqaaiaadMgaaeqaaOGaai4oaaaa@3990@ des exemples communs sont des totaux, Y i = j U i y j , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamywamaaBa aaleaacaWGPbaabeaakiabg2da9maaqababaGaamyEamaaBaaaleaa caWGQbaabeaaaeaacaWGQbGaaGPaVlabgIGiolaaykW7caWGvbWaaS baaWqaaiaadMgaaeqaaaWcbeqdcqGHris5aOGaaiilaaaa@4533@ ou des moyennes, Y ¯ i = Y i / N i . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmywayaara WaaSbaaSqaaiaadMgaaeqaaOGaeyypa0ZaaSGbaeaacaWGzbWaaSba aSqaaiaadMgaaeqaaaGcbaGaamOtamaaBaaaleaacaWGPbaabeaaaa GccaGGUaaaaa@3DD8@ Tel que susmentionné, le vecteur des variables auxiliaires peut être différent de celui qui est utilisé dans l’estimation directe et désigné par z . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCOEaiaac6 caaaa@37AC@ Le modèle au niveau du domaine peut s’exprimer à l’aide de deux équations.

La première équation, communément appelée le modèle d’échantillonnage, est obtenue comme suit :

θ ^ i = θ i + e i ( 3.1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiUdeNbaK aadaWgaaWcbaGaamyAaaqabaGccqGH9aqpcqaH4oqCdaWgaaWcbaGa amyAaaqabaGccqGHRaWkcaWGLbWaaSbaaSqaaiaadMgaaeqaaOGaaG zbVlaaywW7caaMf8UaaGzbVlaaywW7caGGOaGaaG4maiaac6cacaaI XaGaaiykaaaa@4AF9@

et exprime l’estimation directe θ ^ i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiUdeNbaK aadaWgaaWcbaGaamyAaaqabaaaaa@38D7@ en fonction du paramètre inconnu θ i , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiUde3aaS baaSqaaiaadMgaaeqaaOGaaiilaaaa@3981@ plus une erreur aléatoire e i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyzamaaBa aaleaacaWGPbaabeaaaaa@37FB@ attribuable à l’échantillonnage. Les erreurs d’échantillonnage e i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyzamaaBa aaleaacaWGPbaabeaaaaa@37FB@ sont des erreurs indépendantes et identiquement distribuées de moyenne 0 et de variance ψ i , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiYdK3aaS baaSqaaiaadMgaaeqaaOGaaiilaaaa@3999@ c’est-à-dire E p ( e i | θ i ) = 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyramaaBa aaleaacaWGWbaabeaakmaabmaabaGaamyzamaaBaaaleaacaWGPbaa beaakiaaykW7daabbaqaaiaaykW7cqaH4oqCdaWgaaWcbaGaamyAaa qabaaakiaawEa7aaGaayjkaiaawMcaaiabg2da9iaaicdaaaa@44C7@ et V p ( e i | θ i ) = ψ i , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOvamaaBa aaleaacaWGWbaabeaakmaabmaabaGaamyzamaaBaaaleaacaWGPbaa beaakiaaykW7daabbaqaaiaaykW7cqaH4oqCdaWgaaWcbaGaamyAaa qabaaakiaawEa7aaGaayjkaiaawMcaaiabg2da9iabeI8a5naaBaaa leaacaWGPbaabeaakiaacYcaaaa@47C0@ p MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCaaaa@36EC@ désigne l’espérance relative au plan d’échantillonnage. À noter que ψ i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiYdK3aaS baaSqaaiaadMgaaeqaaaaa@38DF@ est aussi la variance sous le plan de θ ^ i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiUdeNbaK aadaWgaaWcbaGaamyAaaqabaaaaa@38D7@ et qu’il est habituellement inconnu.

La seconde équation, connue comme le modèle de lien, est obtenue comme suit :

θ i = z i T β + b i v i ( 3.2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiUde3aaS baaSqaaiaadMgaaeqaaOGaeyypa0JaaCOEamaaDaaaleaacaWGPbaa baGaamivaaaakiaahk7acqGHRaWkcaWGIbWaaSbaaSqaaiaadMgaae qaaOGaamODamaaBaaaleaacaWGPbaabeaakiaaywW7caaMf8UaaGzb VlaaywW7caaMf8UaaiikaiaaiodacaGGUaGaaGOmaiaacMcaaaa@4E6B@

et exprime le paramètre θ i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiUde3aaS baaSqaaiaadMgaaeqaaaaa@38C7@ comme un effet fixe z i T β , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCOEamaaDa aaleaacaWGPbaabaGaamivaaaakiaahk7acaGGSaaaaa@3AE6@ plus un effet aléatoire v i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODamaaBa aaleaacaWGPbaabeaaaaa@380C@ multiplié par b i . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOyamaaBa aaleaacaWGPbaabeaakiaac6caaaa@38B4@ Dans le système de production, le terme b i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOyamaaBa aaleaacaWGPbaabeaaaaa@37F8@ a une valeur par défaut de un, mais l’utilisateur peut le préciser pour contrôler les erreurs hétéroscédastiques ou l’impact des observations influentes. Les effets aléatoires v i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODamaaBa aaleaacaWGPbaabeaaaaa@380C@ sont indépendants et identiquement distribués de variance moyenne 0 et de variance du modèle inconnue σ v 2 , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aa0 baaSqaaiaadAhaaeaacaaIYaaaaOGaaiilaaaa@3A58@ c’est-à-dire E m ( v i ) = 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyramaaBa aaleaacaWGTbaabeaakmaabmaabaGaamODamaaBaaaleaacaWGPbaa beaaaOGaayjkaiaawMcaaiabg2da9iaaicdaaaa@3D51@ et V m ( v i ) = σ v 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOvamaaBa aaleaacaWGTbaabeaakmaabmaabaGaamODamaaBaaaleaacaWGPbaa beaaaOGaayjkaiaawMcaaiabg2da9iabeo8aZnaaDaaaleaacaWG2b aabaGaaGOmaaaaaaa@404F@ E m MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyramaaBa aaleaacaWGTbaabeaaaaa@37DF@ désigne l’espérance du modèle et V m , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOvamaaBa aaleaacaWGTbaabeaakiaacYcaaaa@38AA@ la variance du modèle. Les erreurs aléatoires e i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyzamaaBa aaleaacaWGPbaabeaaaaa@37FB@ sont indépendantes des effets aléatoires v i . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODamaaBa aaleaacaWGPbaabeaakiaac6caaaa@38C8@ La combinaison du modèle d’échantillonnage et du modèle de lien donne un seul modèle linéaire mixte généralisé (MLMG), obtenu comme ceci :

θ ^ i = z i T β + b i v i + e i . ( 3.3 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiUdeNbaK aadaWgaaWcbaGaamyAaaqabaGccqGH9aqpcaWH6bWaa0baaSqaaiaa dMgaaeaacaWGubaaaOGaaCOSdiabgUcaRiaadkgadaWgaaWcbaGaam yAaaqabaGccaWG2bWaaSbaaSqaaiaadMgaaeqaaOGaey4kaSIaamyz amaaBaaaleaacaWGPbaabeaakiaac6cacaaMf8UaaGzbVlaaywW7ca aMf8UaaGzbVlaacIcacaaIZaGaaiOlaiaaiodacaGGPaaaaa@521E@

Dans le modèle de Fay-Herriot (3.3), nous constatons que E m p ( θ ^ i ) = z i T β MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyramaaBa aaleaacaWGTbGaamiCaaqabaGcdaqadaqaaiqbeI7aXzaajaWaaSba aSqaaiaadMgaaeqaaaGccaGLOaGaayzkaaGaeyypa0JaaCOEamaaDa aaleaacaWGPbaabaGaamivaaaakiaahk7aaaa@4296@ et V m p ( θ ^ i ) = b i 2 σ v 2 + ψ ˜ i , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOvamaaBa aaleaacaWGTbGaamiCaaqabaGcdaqadaqaaiqbeI7aXzaajaWaaSba aSqaaiaadMgaaeqaaaGccaGLOaGaayzkaaGaeyypa0JaamOyamaaDa aaleaacaWGPbaabaGaaGOmaaaakiabeo8aZnaaDaaaleaacaWG2baa baGaaGOmaaaakiabgUcaRiqbeI8a5zaaiaWaaSbaaSqaaiaadMgaae qaaOGaaiilaaaa@4974@ ψ ˜ i = E m ( ψ i ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiYdKNbaG aadaWgaaWcbaGaamyAaaqabaGccqGH9aqpcaWGfbWaaSbaaSqaaiaa d2gaaeqaaOWaaeWaaeaacqaHipqEdaWgaaWcbaGaamyAaaqabaaaki aawIcacaGLPaaaaaa@406B@ est la variance sous le plan lisse de θ ^ i . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiUdeNbaK aadaWgaaWcbaGaamyAaaqabaGccaGGUaaaaa@3993@ En général, nous ne pouvons pas traiter ψ i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiYdK3aaS baaSqaaiaadMgaaeqaaaaa@38DF@ comme étant fixe, parce qu’il n’est pas strictement une fonction des données auxiliaires. Si les σ v 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aa0 baaSqaaiaadAhaaeaacaaIYaaaaaaa@399E@ et ψ ˜ i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiYdKNbaG aadaWgaaWcbaGaamyAaaqabaaaaa@38EE@ sont connus, la solution du MLMG donne la meilleure estimation linéaire sans biais empirique (BLUP), θ ˜ i BLUP MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiUdeNbaG aadaqhaaWcbaGaamyAaaqaaiaabkeacaqGmbGaaeyvaiaabcfaaaaa aa@3C16@

θ ˜ i BLUP = { γ i θ ^ i + ( 1 γ i ) z i T β ˜ pour i A z i T β ˜ pour i A ¯ ( 3.4 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiUdeNbaG aadaqhaaWcbaGaamyAaaqaaiaabkeacaqGmbGaaeyvaiaabcfaaaGc cqGH9aqpdaGabaqaauaabaqaciaaaeaacqaHZoWzdaWgaaWcbaGaam yAaaqabaGccuaH4oqCgaqcamaaBaaaleaacaWGPbaabeaakiabgUca RmaabmaabaGaaGymaiabgkHiTiabeo7aNnaaBaaaleaacaWGPbaabe aaaOGaayjkaiaawMcaaiaahQhadaqhaaWcbaGaamyAaaqaaiaadsfa aaGcceWHYoGbaGaaaeaacaqGWbGaae4BaiaabwhacaqGYbGaaGjbVl aaykW7caWGPbGaeyicI4SaamyqaaqaaiaahQhadaqhaaWcbaGaamyA aaqaaiaadsfaaaGcceWHYoGbaGaaaeaacaqGWbGaae4Baiaabwhaca qGYbGaaGjbVlaaykW7caWGPbGaeyicI4SabmyqayaaraaccmGae8hi aacaaaGaay5EaaGaaGzbVlaaywW7caaMf8UaaGzbVlaaywW7caGGOa GaaG4maiaac6cacaaI0aGaaiykaaaa@73F7@

γ i = ( b i 2 σ v 2 ) / ( ψ ˜ i + b i 2 σ v 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4SdC2aaS baaSqaaiaadMgaaeqaaOGaeyypa0ZaaSGbaeaadaqadaqaaiaadkga daqhaaWcbaGaamyAaaqaaiaaikdaaaGccqaHdpWCdaqhaaWcbaGaam ODaaqaaiaaikdaaaaakiaawIcacaGLPaaaaeaadaqadaqaaiqbeI8a 5zaaiaWaaSbaaSqaaiaadMgaaeqaaOGaey4kaSIaamOyamaaDaaale aacaWGPbaabaGaaGOmaaaakiabeo8aZnaaDaaaleaacaWG2baabaGa aGOmaaaaaOGaayjkaiaawMcaaaaaaaa@4DC5@ et β ˜ = ( i A z i z i T / ( ψ ˜ i + b i 2 σ v 2 ) ) 1 i A z i θ ^ i / ( ψ ˜ i + b i 2 σ v 2 ) . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabCOSdyaaia Gaeyypa0ZaaeWaaeaadaWcgaqaamaaqababaGaaCOEamaaBaaaleaa caWGPbaabeaakiaahQhadaqhaaWcbaGaamyAaaqaaiaadsfaaaaaba GaamyAaiabgIGiolaadgeaaeqaniabggHiLdaakeaadaqadaqaaiqb eI8a5zaaiaWaaSbaaSqaaiaadMgaaeqaaOGaey4kaSIaamOyamaaDa aaleaacaWGPbaabaGaaGOmaaaakiabeo8aZnaaDaaaleaacaWG2baa baGaaGOmaaaaaOGaayjkaiaawMcaaaaaaiaawIcacaGLPaaadaahaa WcbeqaaiabgkHiTiaaigdaaaGcdaWcgaqaamaaqababaGaaCOEamaa BaaaleaacaWGPbaabeaakiqbeI7aXzaajaWaaSbaaSqaaiaadMgaae qaaaqaaiaadMgacqGHiiIZcaWGbbaabeqdcqGHris5aaGcbaWaaeWa aeaacuaHipqEgaacamaaBaaaleaacaWGPbaabeaakiabgUcaRiaadk gadaqhaaWcbaGaamyAaaqaaiaaikdaaaGccqaHdpWCdaqhaaWcbaGa amODaaqaaiaaikdaaaaakiaawIcacaGLPaaaaaGaaiOlaaaa@68B3@

Il existe quatre procédures récursives pour l’estimation de σ v 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aa0 baaSqaaiaadAhaaeaacaaIYaaaaaaa@399E@ et β MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCOSdaaa@3735@ dans le système de production. Les trois premières supposent que ψ ˜ i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiYdKNbaG aadaWgaaWcbaGaamyAaaqabaaaaa@38EE@ est connu qu’une version lissée est disponible (voir les détails dans la section suivante). Selon cette hypothèse, les composantes de la variance peuvent être calculées au moyen de la procédure de Fay-Herriot (FH) décrite dans Fay et Herriot (1979), du maximum de vraisemblance restreint (REML) ou de la maximisation de la densité corrigée (ADM) attribuée à Li et Lahiri (2010). La quatrième procédure, WF, attribuée à Wang et Fuller (2003), suppose que nous estimons ψ i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiYdK3aaS baaSqaaiaadMgaaeqaaaaa@38DF@ à l’aide de ψ ^ i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiYdKNbaK aadaWgaaWcbaGaamyAaaqabaaaaa@38EF@ en sachant que n i 2. MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBamaaBa aaleaacaWGPbaabeaakiabgwMiZkaaikdacaGGUaaaaa@3B42@ La procédure WF ne requiert aucun lissage des valeurs estimées de ψ ^ i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiYdKNbaK aadaWgaaWcbaGaamyAaaqabaaaaa@38EF@ avant l’estimation de σ v 2 . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aa0 baaSqaaiaadAhaaeaacaaIYaaaaOGaaiOlaaaa@3A5A@ Wang et Fuller (2003) ont effectué des simulations où n i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBamaaBa aaleaacaWGPbaabeaaaaa@3804@ allait de 9 à 36 et constaté que leur procédure donnait des estimations raisonnables de θ i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiUde3aaS baaSqaaiaadMgaaeqaaaaa@38C7@ et de son erreur quadratique moyenne estimée.

La principale différence entre ces quatre procédures est la façon dont les σ v 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aa0 baaSqaaiaadAhaaeaacaaIYaaaaaaa@399E@ sont calculés. Ils reposent tous sur un algorithme de notation itératif qui donne σ ^ v 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafq4WdmNbaK aadaqhaaWcbaGaamODaaqaaiaaikdaaaaaaa@39AE@ comme une estimation de la variance du modèle σ v 2 . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aa0 baaSqaaiaadAhaaeaacaaIYaaaaOGaaiOlaaaa@3A5A@ Les procédures FH, REML et WF peuvent donner des σ ^ v 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafq4WdmNbaK aadaqhaaWcbaGaamODaaqaaiaaikdaaaaaaa@39AE@ qui sont inférieurs à zéro. Si cela se produit, les σ ^ v 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafq4WdmNbaK aadaqhaaWcbaGaamODaaqaaiaaikdaaaaaaa@39AE@ sont établis à zéro pour les procédures FH et REML. La troncation de σ v 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aa0 baaSqaaiaadAhaaeaacaaIYaaaaaaa@399E@ estimé à zéro a pour inconvénient de rendre l’estimateur sur petits domaines synthétique pour tous les domaines. Li et Lahiri (2010) ont proposé l’ADM afin de régler le problème des σ ^ v 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafq4WdmNbaK aadaqhaaWcbaGaamODaaqaaiaaikdaaaaaaa@39AE@ négatifs en maximisant une probabilité corrigée définie comme le produit de la variance du modèle et d’une probabilité standard. Bien que la méthode de l’ADM donne toujours une solution positive pour σ v 2 , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aa0 baaSqaaiaadAhaaeaacaaIYaaaaOGaaiilaaaa@3A58@ il faudrait l’utiliser avec prudence parce qu’elle surestime la variance du modèle. Les procédures REML, FH et ADM ont recours aux valeurs lissées des valeurs estimées ψ ^ i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiYdKNbaK aadaWgaaWcbaGaamyAaaqabaaaaa@38EF@ obtenues à partir de l’échantillon ou d’une estimation fournie par l’utilisateur. Pour la procédure WF, si σ ^ v 2 < 0 , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafq4WdmNbaK aadaqhaaWcbaGaamODaaqaaiaaikdaaaGccqGH8aapcaaIWaGaaiil aaaa@3C26@ Wang et Fuller (2003) ont suggéré de déterminer σ ^ v 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafq4WdmNbaK aadaqhaaWcbaGaamODaaqaaiaaikdaaaaaaa@39AE@ comme 0,5   V ^ ( σ ^ v 2 ) , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeimaiaabY cacaqG1aGaaGPaVpaakaaabaGaaeiiaiqadAfagaqcamaabmaabaGa fq4WdmNbaKaadaqhaaWcbaGaamODaaqaaiaaikdaaaaakiaawIcaca GLPaaaaSqabaGccaGGSaaaaa@4149@

V ^ ( σ ^ v 2 ) = i A 2 κ i 2 [ ( ψ ^ i + b i 2 σ ^ v 2 ) 2 + ( ψ ^ i ) 2 ( n i 1 ) ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOvayaaja WaaeWaaeaacuaHdpWCgaqcamaaDaaaleaacaWG2baabaGaaGOmaaaa aOGaayjkaiaawMcaaiabg2da9maaqafabaGaaGOmaiabeQ7aRnaaDa aaleaacaWGPbaabaGaaGOmaaaakmaadmaabaWaaeWaaeaacuaHipqE gaqcamaaBaaaleaacaWGPbaabeaakiabgUcaRiaadkgadaqhaaWcba GaamyAaaqaaiaaikdaaaGccuaHdpWCgaqcamaaDaaaleaacaWG2baa baGaaGOmaaaaaOGaayjkaiaawMcaamaaCaaaleqabaGaaGOmaaaaki abgUcaRmaalaaabaWaaeWaaeaacuaHipqEgaqcamaaBaaaleaacaWG PbaabeaaaOGaayjkaiaawMcaamaaCaaaleqabaGaaGOmaaaaaOqaam aabmaabaGaamOBamaaBaaaleaacaWGPbaabeaakiabgkHiTiaaigda aiaawIcacaGLPaaaaaaacaGLBbGaayzxaaaaleaacaWGPbGaeyicI4 Saamyqaaqab0GaeyyeIuoaaaa@616E@

et

κ i = [ b i 2 σ ^ v 2 + ( n i + 1 ) ( n i 1 ) ψ ^ i ] 1 i A [ b i 2 σ ^ v 2 + ( n i + 1 ) ( n i 1 ) ψ ^ i ] 1 . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOUdS2aaS baaSqaaiaadMgaaeqaaOGaeyypa0ZaaSaaaeaadaWadaqaaiaadkga daqhaaWcbaGaamyAaaqaaiaaikdaaaGccuaHdpWCgaqcamaaDaaale aacaWG2baabaGaaGOmaaaakiabgUcaRmaalaaabaWaaeWaaeaacaWG UbWaaSbaaSqaaiaadMgaaeqaaOGaey4kaSIaaGymaaGaayjkaiaawM caaaqaamaabmaabaGaamOBamaaBaaaleaacaWGPbaabeaakiabgkHi TiaaigdaaiaawIcacaGLPaaaaaGafqiYdKNbaKaadaWgaaWcbaGaam yAaaqabaaakiaawUfacaGLDbaadaahaaWcbeqaaiabgkHiTiaaigda aaaakeaadaaeqaqaamaadmaabaGaamOyamaaDaaaleaacaWGPbaaba GaaGOmaaaakiqbeo8aZzaajaWaa0baaSqaaiaadAhaaeaacaaIYaaa aOGaey4kaSYaaSaaaeaadaqadaqaaiaad6gadaWgaaWcbaGaamyAaa qabaGccqGHRaWkcaaIXaaacaGLOaGaayzkaaaabaWaaeWaaeaacaWG UbWaaSbaaSqaaiaadMgaaeqaaOGaeyOeI0IaaGymaaGaayjkaiaawM caaaaacuaHipqEgaqcamaaBaaaleaacaWGPbaabeaaaOGaay5waiaa w2faamaaCaaaleqabaGaeyOeI0IaaGymaaaaaeaacaWGPbGaeyicI4 Saamyqaaqab0GaeyyeIuoaaaGccaGGUaaaaa@714A@

L’ajout de σ ^ v 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafq4WdmNbaK aadaqhaaWcbaGaamODaaqaaiaaikdaaaaaaa@39AE@ et d’une estimation des ψ ˜ i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiYdKNbaG aadaWgaaWcbaGaamyAaaqabaaaaa@38EE@ à θ ˜ i BLUP , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiUdeNbaG aadaqhaaWcbaGaamyAaaqaaiaabkeacaqGmbGaaeyvaiaabcfaaaGc caGGSaaaaa@3CD0@ défini à l’aide de l’équation (3.4), donne la meilleure estimation linéaire sans biais empirique (EBLUP), θ ^ i EBLUP , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiUdeNbaK aadaqhaaWcbaGaamyAaaqaaiaabweacaqGcbGaaeitaiaabwfacaqG qbaaaOGaaiilaaaa@3D99@ qui est obtenue comme suit :

θ ^ i EBLUP = { γ ^ i θ ^ i + ( 1 γ ^ i ) z i T β ^ pour  i A z i T β ^ pour  i A ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiUdeNbaK aadaqhaaWcbaGaamyAaaqaaiaabweacaqGcbGaaeitaiaabwfacaqG qbaaaOGaeyypa0ZaaiqaaeaafaqaaeGacaaabaGafq4SdCMbaKaada WgaaWcbaGaamyAaaqabaGccuaH4oqCgaqcamaaBaaaleaacaWGPbaa beaakiabgUcaRmaabmaabaGaaGymaiabgkHiTiqbeo7aNzaajaWaaS baaSqaaiaadMgaaeqaaaGccaGLOaGaayzkaaGaaCOEamaaDaaaleaa caWGPbaabaGaamivaaaakiqahk7agaqcaaqaaiaabchacaqGVbGaae yDaiaabkhacaqGGaGaamyAaiabgIGiolaadgeaaeaacaWH6bWaa0ba aSqaaiaadMgaaeaacaWGubaaaOGabCOSdyaajaaabaGaaeiCaiaab+ gacaqG1bGaaeOCaiaabccacaWGPbGaeyicI4SabmyqayaaraaaaaGa ay5Eaaaaaa@63DA@

γ ^ i = ( b i 2 σ ^ v 2 ) / ( ψ ¨ i + b i 2 σ ^ v 2 ) , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafq4SdCMbaK aadaWgaaWcbaGaamyAaaqabaGccqGH9aqpdaWcgaqaamaabmaabaGa amOyamaaDaaaleaacaWGPbaabaGaaGOmaaaakiqbeo8aZzaajaWaa0 baaSqaaiaadAhaaeaacaaIYaaaaaGccaGLOaGaayzkaaaabaWaaeWa aeaacuaHipqEgaWaamaaBaaaleaacaWGPbaabeaakiabgUcaRiaadk gadaqhaaWcbaGaamyAaaqaaiaaikdaaaGccuaHdpWCgaqcamaaDaaa leaacaWG2baabaGaaGOmaaaaaOGaayjkaiaawMcaaaaacaGGSaaaaa@4EA0@ β ^ = ( i A z i z i T / ( ψ ¨ i + b i 2 σ ^ v 2 ) ) 1 i A z i θ ^ i DIR / ( ψ ¨ i + b i 2 σ ^ v 2 ) , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabCOSdyaaja Gaeyypa0ZaaeWaaeaadaWcgaqaamaaqababaGaaCOEamaaBaaaleaa caWGPbaabeaakiaahQhadaqhaaWcbaGaamyAaaqaaiaadsfaaaaaba GaamyAaiabgIGiolaadgeaaeqaniabggHiLdaakeaadaqadaqaaiqb eI8a5zaadaWaaSbaaSqaaiaadMgaaeqaaOGaey4kaSIaamOyamaaDa aaleaacaWGPbaabaGaaGOmaaaakiqbeo8aZzaajaWaa0baaSqaaiaa dAhaaeaacaaIYaaaaaGccaGLOaGaayzkaaaaaaGaayjkaiaawMcaam aaCaaaleqabaGaeyOeI0IaaGymaaaakmaalyaabaWaaabeaeaacaWH 6bWaaSbaaSqaaiaadMgaaeqaaOGafqiUdeNbaKaadaqhaaWcbaGaam yAaaqaaiaabseacaqGjbGaaeOuaaaaaeaacaWGPbGaeyicI4Saamyq aaqab0GaeyyeIuoaaOqaamaabmaabaGafqiYdKNbamaadaWgaaWcba GaamyAaaqabaGccqGHRaWkcaWGIbWaa0baaSqaaiaadMgaaeaacaaI YaaaaOGafq4WdmNbaKaadaqhaaWcbaGaamODaaqaaiaaikdaaaaaki aawIcacaGLPaaaaaGaaiilaaaa@6B31@ et ψ ¨ i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiYdKNbam aadaWgaaWcbaGaamyAaaqabaaaaa@38E9@ est choisi selon la procédure employée. Pour les procédures REML, FH et ADM, les ψ ¨ i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiYdKNbam aadaWgaaWcbaGaamyAaaqabaaaaa@38E9@ sont les valeurs lissées des valeurs estimées de ψ ^ i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiYdKNbaK aadaWgaaWcbaGaamyAaaqabaaaaa@38EF@ obtenues à partir de l’échantillon ou d’une estimation fournie par l’utilisateur. Pour la procédure WF, nous disons que ψ ¨ i = ψ ^ i . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiYdKNbam aadaWgaaWcbaGaamyAaaqabaGccqGH9aqpcuaHipqEgaqcamaaBaaa leaacaWGPbaabeaakiaac6caaaa@3DAD@ Si la variance du modèle estimée b i 2 σ ^ v 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOyamaaDa aaleaacaWGPbaabaGaaGOmaaaakiqbeo8aZzaajaWaa0baaSqaaiaa dAhaaeaacaaIYaaaaaaa@3C76@ est relativement petite comparativement à ψ ¨ i , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiYdKNbam aadaWgaaWcbaGaamyAaaqabaGccaGGSaaaaa@39A3@ alors γ ^ i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafq4SdCMbaK aadaWgaaWcbaGaamyAaaqabaaaaa@38C8@ est petit et plus de poids est attaché à l’estimateur synthétique z i T β ^ . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCOEamaaDa aaleaacaWGPbaabaGaamivaaaakiqahk7agaqcaiaac6caaaa@3AF8@ De même, plus de poids est attaché à l’estimateur direct, θ ^ i , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiUdeNbaK aadaWgaaWcbaGaamyAaaqabaGccaGGSaaaaa@3991@ si la variance sous le plan ψ ¨ i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiYdKNbam aadaWgaaWcbaGaamyAaaqabaaaaa@38E9@ est relativement petite.

Les détails des calculs requis se trouvent dans les spécifications de la méthodologie applicable au système de production décrites dans Estevao et coll. (2015).

3.1  Estimation de la variance sous le plan lisse

La variance sous le plan, ψ i , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiYdK3aaS baaSqaaiaadMgaaeqaaOGaaiilaaaa@3999@ pourrait être utilisée comme estimateur de la variance sous le plan lisse ψ ˜ i = E m ( ψ i ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiYdKNbaG aadaWgaaWcbaGaamyAaaqabaGccqGH9aqpcaWGfbWaaSbaaSqaaiaa d2gaaeqaaOWaaeWaaeaacqaHipqEdaWgaaWcbaGaamyAaaqabaaaki aawIcacaGLPaaaaaa@406A@ si elle était connue. Dans la plupart des cas, elle est inconnue. Pour contourner ce problème, on suppose qu’un estimateur de variance sous le plan sans biais ψ ^ i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiYdKNbaK aadaWgaaWcbaGaamyAaaqabaaaaa@38EF@ de ψ i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiYdK3aaS baaSqaaiaadMgaaeqaaaaa@38DF@ est disponible, c’est-à-dire que E p ( ψ ^ i ) = ψ i . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyramaaBa aaleaacaWGWbaabeaakmaabmaabaGafqiYdKNbaKaadaWgaaWcbaGa amyAaaqabaaakiaawIcacaGLPaaacqGH9aqpcqaHipqEdaWgaaWcba GaamyAaaqabaGccaGGUaaaaa@4121@ Selon cette hypothèse, nous avons ceci :

E m p ( ψ ^ i ) = E m ( ψ i ) = ψ ˜ i . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyramaaBa aaleaacaWGTbGaamiCaaqabaGcdaqadaqaaiqbeI8a5zaajaWaaSba aSqaaiaadMgaaeqaaaGccaGLOaGaayzkaaGaeyypa0JaamyramaaBa aaleaacaWGTbaabeaakmaabmaabaGaeqiYdK3aaSbaaSqaaiaadMga aeqaaaGccaGLOaGaayzkaaGaeyypa0JafqiYdKNbaGaadaWgaaWcba GaamyAaaqabaGccaGGUaaaaa@4994@

Un estimateur simple sans biais de la variance sous le plan lisse ψ ˜ i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiYdKNbaG aadaWgaaWcbaGaamyAaaqabaaaaa@38EE@ est ψ ^ i . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiYdKNbaK aadaWgaaWcbaGaamyAaaqabaGccaGGUaaaaa@39AB@ Toutefois, ψ ^ i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiYdKNbaK aadaWgaaWcbaGaamyAaaqabaaaaa@38EF@ peut être assez instable lorsque la taille de l’échantillon dans le domaine i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAaaaa@36E5@ est petite. On obtient un estimateur plus efficace en modélisant ψ ^ i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiYdKNbaK aadaWgaaWcbaGaamyAaaqabaaaaa@38EF@ avec z i . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCOEamaaBa aaleaacaWGPbaabeaakiaac6caaaa@38D0@ Dick (1995) et Rivest et Belmonte (2000) ont envisagé des modèles de lissage obtenus comme ceci :

log ( ψ ^ i ) = x i T α + ε i , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciiBaiaac+ gacaGGNbWaaeWaaeaacuaHipqEgaqcamaaBaaaleaacaWGPbaabeaa aOGaayjkaiaawMcaaiabg2da9iaahIhadaqhaaWcbaGaamyAaaqaai aadsfaaaGccaWHXoGaey4kaSIaeqyTdu2aaSbaaSqaaiaadMgaaeqa aOGaaiilaaaa@46F0@

x i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCiEamaaBa aaleaacaWGPbaabeaaaaa@3812@ est un vecteur des variables explicatives qui sont des fonctions de z i , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCOEamaaBa aaleaacaWGPbaabeaakiaacYcaaaa@38CE@ α MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCySdaaa@3734@ est un vecteur inconnu des paramètres modèles qu’il faut estimer, et ε i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyTdu2aaS baaSqaaiaadMgaaeqaaaaa@38B8@ est une erreur aléatoire où E m p ( ε i ) = 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyramaaBa aaleaacaWGTbGaamiCaaqabaGcdaqadaqaaiabew7aLnaaBaaaleaa caWGPbaabeaaaOGaayjkaiaawMcaaiabg2da9iaaicdaaaa@3EF2@ et la variance constante σ ε 2 = V m p ( ε i ) . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aa0 baaSqaaiabew7aLbqaaiaaikdaaaGccqGH9aqpcaWGwbWaaSbaaSqa aiaad2gacaWGWbaabeaakmaabmaabaGaeqyTdu2aaSbaaSqaaiaadM gaaeqaaaGccaGLOaGaayzkaaGaaiOlaaaa@4358@ Nous supposons également que les erreurs ε i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyTdu2aaS baaSqaaiaadMgaaeqaaaaa@38B8@ sont identiquement distribuées conditionnellement à z i , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCOEamaaBa aaleaacaWGPbaabeaakiaacYcaaaa@38CE@ i = 1 , , m . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAaiabg2 da9iaaigdacaGGSaGaaGjbVlablAciljaacYcacaaMe8UaamyBaiaa c6caaaa@3FE6@ À partir de ce modèle, nous constatons que :

ψ ˜ i = E m p ( ψ ^ i ) = exp ( x i T α ) Δ , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiYdKNbaG aadaWgaaWcbaGaamyAaaqabaGccqGH9aqpcaWGfbWaaSbaaSqaaiaa d2gacaWGWbaabeaakmaabmaabaGafqiYdKNbaKaadaWgaaWcbaGaam yAaaqabaaakiaawIcacaGLPaaacqGH9aqpciGGLbGaaiiEaiaaccha daqadaqaaiaahIhadaqhaaWcbaGaamyAaaqaaiaadsfaaaGccaWHXo aacaGLOaGaayzkaaGaeuiLdqKaaiilaaaa@4D2B@

Δ = E m p ( exp ( ε i ) ) . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuiLdqKaey ypa0JaamyramaaBaaaleaacaWGTbGaamiCaaqabaGcdaqadaqaaiGa cwgacaGG4bGaaiiCamaabmaabaGaeqyTdu2aaSbaaSqaaiaadMgaae qaaaGccaGLOaGaayzkaaaacaGLOaGaayzkaaGaaiOlaaaa@44B4@ Dick (1995) a estimé ψ ˜ i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiYdKNbaG aadaWgaaWcbaGaamyAaaqabaaaaa@38EE@ en omettant le facteur Δ . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuiLdqKaai Olaaaa@380F@ Rivest et Belmonte (2000) ont estimé Δ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuiLdqeaaa@375D@ en supposant que les erreurs ε i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyTdu2aaS baaSqaaiaadMgaaeqaaaaa@38B8@ étaient normalement distribuées. Cependant, nous avons observé de manière empirique que cet estimateur de Δ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuiLdqeaaa@375D@ était sensible aux écarts par rapport à l’hypothèse de normalité. On évite cette hypothèse en utilisant une méthode des moments (voir Beaumont et Bocci, 2016). Cela donne l’estimateur sans biais de Δ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuiLdqeaaa@375D@ :

Δ ^ ( α ) = i = 1 m ψ ^ i i = 1 m exp ( x i T α ) . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafuiLdqKbaK aadaqadaqaaiaahg7aaiaawIcacaGLPaaacqGH9aqpdaWcaaqaamaa qadabaGafqiYdKNbaKaadaWgaaWcbaGaamyAaaqabaaabaGaamyAai abg2da9iaaigdaaeaacaWGTbaaniabggHiLdaakeaadaaeWaqaaiGa cwgacaGG4bGaaiiCamaabmaabaGaaCiEamaaDaaaleaacaWGPbaaba Gaamivaaaakiaahg7aaiaawIcacaGLPaaaaSqaaiaadMgacqGH9aqp caaIXaaabaGaamyBaaqdcqGHris5aaaakiaac6caaaa@52E1@

Il faut un estimateur α ^ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabCySdyaaja aaaa@3744@ du vecteur des paramètres inconnus du modèle α MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCySdaaa@3734@ pour estimer ψ ˜ i . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiYdKNbaG aadaWgaaWcbaGaamyAaaqabaGccaGGUaaaaa@39AA@ On l’obtient à l’aide de la méthode des moindres carrés ordinaires comme suit :

α ^ = ( i = 1 m x i x i T ) 1 i = 1 m x i log ( ψ ^ i ) . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabCySdyaaja Gaeyypa0ZaaeWaaeaadaaeWbqaaiaahIhadaWgaaWcbaGaamyAaaqa baGccaWH4bWaa0baaSqaaiaadMgaaeaacaWGubaaaaqaaiaadMgacq GH9aqpcaaIXaaabaGaamyBaaqdcqGHris5aaGccaGLOaGaayzkaaWa aWbaaSqabeaacqGHsislcaaIXaaaaOWaaabCaeaacaWH4bWaaSbaaS qaaiaadMgaaeqaaOGaciiBaiaac+gacaGGNbWaaeWaaeaacuaHipqE gaqcamaaBaaaleaacaWGPbaabeaaaOGaayjkaiaawMcaaaWcbaGaam yAaiabg2da9iaaigdaaeaacaWGTbaaniabggHiLdGccaGGUaaaaa@56CC@

On obtient alors l’estimateur ψ ˜ ^ i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiYdKNbaG GbaKaadaWgaaWcbaGaamyAaaqabaaaaa@38FD@ de ψ ˜ i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiYdKNbaG aadaWgaaWcbaGaamyAaaqabaaaaa@38EE@ comme suit :

ψ ˜ ^ i = exp ( x i T α ^ ) Δ ^ ( α ^ ) . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiYdKNbaG GbaKaadaWgaaWcbaGaamyAaaqabaGccqGH9aqpciGGLbGaaiiEaiaa cchadaqadaqaaiaahIhadaqhaaWcbaGaamyAaaqaaiaadsfaaaGcce WHXoGbaKaaaiaawIcacaGLPaaacuqHuoargaqcamaabmaabaGabCyS dyaajaaacaGLOaGaayzkaaGaaiOlaaaa@47BA@

Une belle propriété de ψ ˜ ^ i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiYdKNbaG GbaKaadaWgaaWcbaGaamyAaaqabaaaaa@38FD@ vient du fait que la moyenne de l’estimateur de la variance sous le plan lisse, ψ ˜ ^ i , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiYdKNbaG GbaKaadaWgaaWcbaGaamyAaaqabaGccaGGSaaaaa@39B7@ est égale à la moyenne de l’estimateur de variance directe, ψ ^ i ; MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiYdKNbaK aadaWgaaWcbaGaamyAaaqabaGccaGG7aaaaa@39B8@ c’est-à-dire :

i = 1 m ψ ˜ ^ i m = i = 1 m ψ ^ i m . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaada aeWaqaaiqbeI8a5zaaiyaajaWaaSbaaSqaaiaadMgaaeqaaaqaaiaa dMgacqGH9aqpcaaIXaaabaGaamyBaaqdcqGHris5aaGcbaGaamyBaa aacqGH9aqpdaWcaaqaamaaqadabaGafqiYdKNbaKaadaWgaaWcbaGa amyAaaqabaaabaGaamyAaiabg2da9iaaigdaaeaacaWGTbaaniabgg HiLdaakeaacaWGTbaaaiaac6caaaa@4AF4@

Cela garantit que ψ ˜ ^ i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiYdKNbaG GbaKaadaWgaaWcbaGaamyAaaqabaaaaa@38FD@ ne surestime ou ne sous-estime pas systématiquement ψ ˜ i = E m p ( ψ ^ i ) . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiYdKNbaG aadaWgaaWcbaGaamyAaaqabaGccqGH9aqpcaWGfbWaaSbaaSqaaiaa d2gacaWGWbaabeaakmaabmaabaGafqiYdKNbaKaadaWgaaWcbaGaam yAaaqabaaakiaawIcacaGLPaaacaGGUaaaaa@4222@

3.2  Réconciliation

Si le paramètre d’intérêt θ i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiUde3aaS baaSqaaiaadMgaaeqaaaaa@38C7@ est un total ( θ i = Y i ) , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaacq aH4oqCdaWgaaWcbaGaamyAaaqabaGccqGH9aqpcaWGzbWaaSbaaSqa aiaadMgaaeqaaaGccaGLOaGaayzkaaGaaiilaaaa@3E12@ l’utilisateur peut souhaiter que la somme des estimations sur petits domaines, θ ^ = i A A ¯ θ ^ i EBLUP , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiUdeNbaK aacqGH9aqpdaaeqaqaaiqbeI7aXzaajaWaa0baaSqaaiaadMgaaeaa caqGfbGaaeOqaiaabYeacaqGvbGaaeiuaaaaaeaacaWGPbGaeyicI4 SaamyqaiabgQIiilqadgeagaqeaaqab0GaeyyeIuoakiaacYcaaaa@47F3@ corresponde aux totaux estimés Y ^ = i A Y ^ i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmywayaaja Gaeyypa0ZaaabeaeaaceWGzbGbaKaadaWgaaWcbaGaamyAaaqabaaa baGaamyAaiabgIGiolaadgeaaeqaniabggHiLdaaaa@3F03@ au niveau de l’échantillon général s ; MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4CaiaacU daaaa@37AE@ c’est-à-dire θ ^ = Y ^ . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiUdeNbaK aacqGH9aqpceWGzbGbaKaacaGGUaaaaa@3A63@ Dans le cas d’une moyenne, θ i = Y ¯ i , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiUde3aaS baaSqaaiaadMgaaeqaaOGaeyypa0JabmywayaaraWaaSbaaSqaaiaa dMgaaeqaaOGaaiilaaaa@3CA1@ cette condition de réconciliation devient i A A ¯ N i θ ^ i EBLUP = i A N i θ ^ i , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaabeaeaaca WGobWaaSbaaSqaaiaadMgaaeqaaOGafqiUdeNbaKaadaqhaaWcbaGa amyAaaqaaiaabweacaqGcbGaaeitaiaabwfacaqGqbaaaaqaaiaadM gacqGHiiIZcaWGbbGaeyOkIGSabmyqayaaraaabeqdcqGHris5aOGa eyypa0ZaaabeaeaacaWGobWaaSbaaSqaaiaadMgaaeqaaOGafqiUde NbaKaadaWgaaWcbaGaamyAaaqabaaabaGaamyAaiabgIGiolaadgea aeqaniabggHiLdGccaGGSaaaaa@5215@ θ ^ i = Y ¯ ^ i . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiUdeNbaK aadaWgaaWcbaGaamyAaaqabaGccqGH9aqpceWGzbGbaeHbaKaadaWg aaWcbaGaamyAaaqabaGccaGGUaaaaa@3CC2@

Dans le système de production, deux méthodes permettent d’assurer la réconciliation des estimations sur petits domaines au niveau du domaine. La première repose sur un rajustement de la différence et la seconde, sur un vecteur enrichi. Elles sont valides pour toute méthode servant à calculer θ ^ i EBLUP MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiUdeNbaK aadaqhaaWcbaGaamyAaaqaaiaabweacaqGcbGaaeitaiaabwfacaqG qbaaaaaa@3CDF@ ou pour déterminer si l’estimation de la variance ψ ¨ i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiYdKNbam aadaWgaaWcbaGaamyAaaqabaaaaa@38E9@ a été lissée. La réconciliation fondée sur un rajustement de la différence est une adaptation de la réconciliation présentée dans Battese et coll. (1988). La réconciliation fondée sur un vecteur enrichi est attribué à Wang, Fuller et Qu (2008).

Rajustement de la différence : Pour cette méthode, l’estimateur θ ^ i EBLUP MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiUdeNbaK aadaqhaaWcbaGaamyAaaqaaiaabweacaqGcbGaaeitaiaabwfacaqG qbaaaaaa@3CDF@ est ajusté uniquement pour les domaines dont la taille d’échantillon réalisé n i 1 , i A MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBamaaBa aaleaacaWGPbaabeaakiabgwMiZkaaigdacaGGSaGaaGjbVlaaykW7 caWGPbGaeyicI4Saamyqaaaa@418F@ et les estimations synthétiques z i T β ^ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCOEamaaDa aaleaacaWGPbaabaGaamivaaaakiqahk7agaqcaaaa@3A46@ pour i A ¯ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAaiabgI Giolqadgeagaqeaaaa@3947@ restent telles quelles. L’estimateur ainsi réconcilié est obtenu à l’aide de θ ^ i EBLUP , b MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiUdeNbaK aadaqhaaWcbaGaamyAaaqaaiaabweacaqGcbGaaeitaiaabwfacaqG qbGaaiilaiaaysW7caWGIbaaaaaa@4003@ et est défini comme suit :

θ ^ i EBLUP , b = { θ ^ i EBLUP + α i ( θ ^ * d A ω d θ ^ d EBLUP ) pour i A z i T β ^ pour i A ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiUdeNbaK aadaqhaaWcbaGaamyAaaqaaiaabweacaqGcbGaaeitaiaabwfacaqG qbGaaiilaiaaysW7caWGIbaaaOGaeyypa0ZaaiqaaeaafaqaaeGaca aabaGafqiUdeNbaKaadaqhaaWcbaGaamyAaaqaaiaabweacaqGcbGa aeitaiaabwfacaqGqbaaaOGaey4kaSIaeqySde2aaSbaaSqaaiaadM gaaeqaaOWaaeWaaeaacuaH4oqCgaqcamaaCaaaleqabaGaaiOkaaaa kiabgkHiTmaaqafabaGaeqyYdC3aaSbaaSqaaiaadsgaaeqaaOGafq iUdeNbaKaadaqhaaWcbaGaamizaaqaaiaabweacaqGcbGaaeitaiaa bwfacaqGqbaaaaqaaiaadsgacqGHiiIZcaWGbbaabeqdcqGHris5aa GccaGLOaGaayzkaaaabaGaaeiCaiaab+gacaqG1bGaaeOCaiaaysW7 caaMc8UaamyAaiabgIGiolaadgeaaeaacaWH6bWaa0baaSqaaiaadM gaaeaacaWGubaaaOGabCOSdyaajaaabaGaaeiCaiaab+gacaqG1bGa aeOCaiaaysW7caaMc8UaamyAaiabgIGiolqadgeagaqeaaaaaiaawU haaaaa@79C2@

α i = { i U A ω i 2 ( ψ ¨ i + b i 2 σ ^ v 2 ) } 1 ω i ( ψ ¨ i + b i 2 σ ^ v 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqefi0BVTwBH5 gipXgzGmfD5XwzaGqbaKqzaeGaa8hiaOGaeqySde2aaSbaaSqaaiaa dMgaaeqaaOGaeyypa0ZaaiWaaeaadaaeqaqaaiabeM8a3naaDaaale aacaWGPbaabaGaaGOmaaaakmaabmaabaGafqiYdKNbamaadaWgaaWc baGaamyAaaqabaGccqGHRaWkcaWGIbWaa0baaSqaaiaadMgaaeaaca aIYaaaaOGafq4WdmNbaKaadaqhaaWcbaGaamODaaqaaiaaikdaaaaa kiaawIcacaGLPaaaaSqaaiaadMgacqGHiiIZcaWGvbWaaSbaaWqaai aadgeaaeqaaaWcbeqdcqGHris5aaGccaGL7bGaayzFaaWaaWbaaSqa beaacqGHsislcaaIXaaaaOGaeqyYdC3aaSbaaSqaaiaadMgaaeqaaO WaaeWaaeaacuaHipqEgaWaamaaBaaaleaacaWGPbaabeaakiabgUca RiaadkgadaqhaaWcbaGaamyAaaqaaiaaikdaaaGccuaHdpWCgaqcam aaDaaaleaacaWG2baabaGaaGOmaaaaaOGaayjkaiaawMcaaGGadKqz aeGae4hiaacaaa@6B3A@ pour i A , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAaiabgI GiolaadgeacaGGSaaaaa@39DF@ ω i = 1 , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyYdC3aaS baaSqaaiaadMgaaeqaaOGaeyypa0JaaGymaiaacYcaaaa@3B59@ si la réconciliation est pour un total, et ω i = N i / N , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyYdC3aaS baaSqaaiaadMgaaeqaaOGaeyypa0ZaaSGbaeaacaWGobWaaSbaaSqa aiaadMgaaeqaaaGcbaGaamOtaaaacaGGSaaaaa@3D7E@ si la réconciliation est pour la moyenne. L’estimateur θ ^ * MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiUdeNbaK aadaahaaWcbeqaaiaacQcaaaaaaa@3898@ est une valeur fournie par l’utilisateur qui représente le total ou la moyenne des y MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEaGqaai aa=1kaaaa@382C@ -valeurs de population U . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyvaiaac6 caaaa@3783@ La réconciliation garantit que i A A ¯ ω i θ ^ i EBLUP , b = θ ^ * . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaabeaeaacq aHjpWDdaWgaaWcbaGaamyAaaqabaGccuaH4oqCgaqcamaaDaaaleaa caWGPbaabaGaaeyraiaabkeacaqGmbGaaeyvaiaabcfacaGGSaGaaG jbVlaadkgaaaaabaGaamyAaiabgIGiolaadgeacqGHQicYceWGbbGb aebaaeqaniabggHiLdGccqGH9aqpcuaH4oqCgaqcamaaCaaaleqaba GaaiOkaaaakiaac6caaaa@4EEF@

Vecteur enrichi : Le vecteur z i T MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCOEamaaDa aaleaacaWGPbaabaGaamivaaaaaaa@38EE@ est enrichi à l’aide de ω i ψ ¨ i , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyYdC3aaS baaSqaaiaadMgaaeqaaOGafqiYdKNbamaadaWgaaWcbaGaamyAaaqa baGccaGGSaaaaa@3C94@ pour former z i * T = ( z i T , ω i ψ ¨ i ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCOEamaaDa aaleaacaWGPbaabaGaaiOkaiaadsfaaaGccqGH9aqpdaqadaqaaiaa hQhadaqhaaWcbaGaamyAaaqaaiaadsfaaaGccaaMb8Uaaiilaiaays W7cqaHjpWDdaWgaaWcbaGaamyAaaqabaGccuaHipqEgaWaamaaBaaa leaacaWGPbaabeaaaOGaayjkaiaawMcaaaaa@48EA@ ω i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyYdC3aaS baaSqaaiaadMgaaeqaaaaa@38DE@ et ψ ¨ i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiYdKNbam aadaWgaaWcbaGaamyAaaqabaaaaa@38E9@ sont définis comme précédemment. L’équation du modèle linéaire mixte généralisé (MLMG) enrichi qui en découle est obtenue comme ceci :

θ ^ i = z i * T β * + b i v i * + e i ( 3.5 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiUdeNbaK aadaWgaaWcbaGaamyAaaqabaGccqGH9aqpcaWH6bWaa0baaSqaaiaa dMgaaeaacaGGQaGaamivaaaakiaahk7adaahaaWcbeqaaiaabQcaaa accmGccqWFGaaicqGHRaWkcaWGIbWaaSbaaSqaaiaadMgaaeqaaOGa amODamaaDaaaleaacaWGPbaabaGaaiOkaaaakiabgUcaRiaadwgada WgaaWcbaGaamyAaaqabaGccaaMf8UaaGzbVlaaywW7caaMf8UaaGzb VlaacIcacaaIZaGaaiOlaiaaiwdacaGGPaaaaa@5481@

E m ( v i * ) = 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyramaaBa aaleaacaWGTbaabeaakmaabmaabaGaamODamaaDaaaleaacaWGPbaa baGaaiOkaaaaaOGaayjkaiaawMcaaiabg2da9iaaicdaaaa@3E00@ et V m ( v i * ) = σ v * 2 . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOvamaaBa aaleaacaWGTbaabeaakmaabmaabaGaamODamaaDaaaleaacaWGPbaa baGaaiOkaaaaaOGaayjkaiaawMcaaiabg2da9iabeo8aZnaaDaaale aacaWG2baabaGaaiOkaiaaikdaaaGccaGGUaaaaa@4268@ Les estimations de β * MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCOSdmaaCa aaleqabaGaaiOkaaaaaaa@3810@ et σ v * 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aa0 baaSqaaiaadAhaaeaacaGGQaGaaGOmaaaaaaa@3A4C@ sont une fois de plus résolues de manière récursive pour les quatre procédures EBLUP que nous désignons comme θ ^ i EBLUP* . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiUdeNbaK aadaqhaaWcbaGaamyAaaqaaiaabweacaqGcbGaaeitaiaabwfacaqG qbGaaeOkaaaakiaac6caaaa@3E48@

L’estimateur réconcilié θ ^ i EBLUP * , b MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiUdeNbaK aadaqhaaWcbaGaamyAaaqaaiaabweacaqGcbGaaeitaiaabwfacaqG qbGaaiOkaiaacYcacaaMe8UaamOyaaaaaaa@40B1@ qui en découle est obtenu comme ceci :

θ ^ i EBLUP * , b = { γ ^ i * θ ^ i EBLUP * + ( 1 γ ^ i * ) z i * T β ^ * pour i A z i * T β ^ * pour i A ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiUdeNbaK aadaqhaaWcbaGaamyAaaqaaiaabweacaqGcbGaaeitaiaabwfacaqG qbWaaWbaaWqabeaacaGGQaaaaSGaaGzaVlaacYcacaaMe8UaamOyaa aakiabg2da9maaceaabaqbaeaabiGaaaqaaiqbeo7aNzaajaWaa0ba aSqaaiaadMgaaeaacaGGQaaaaOGafqiUdeNbaKaadaqhaaWcbaGaam yAaaqaaiaabweacaqGcbGaaeitaiaabwfacaqGqbWaaWbaaWqabeaa caGGQaaaaaaakiabgUcaRmaabmaabaGaaGymaiabgkHiTiqbeo7aNz aajaWaa0baaSqaaiaadMgaaeaacaGGQaaaaaGccaGLOaGaayzkaaGa aCOEamaaDaaaleaacaWGPbaabaGaaiOkaiaadsfaaaGcceWHYoGbaK aadaahaaWcbeqaaiaabQcaaaaakeaacaqGWbGaae4BaiaabwhacaqG YbGaaGjbVlaaykW7caWGPbGaeyicI4SaamyqaaqaaiaahQhadaqhaa WcbaGaamyAaaqaaiaacQcacaWGubaaaGGadOGae8hiaaIabCOSdyaa jaWaaWbaaSqabeaacaqGQaaaaaGcbaGaaeiCaiaab+gacaqG1bGaae OCaiaaysW7caaMc8UaamyAaiabgIGiolqadgeagaqeaaaaaiaawUha aaaa@7891@

γ ^ i * = ( b i 2 σ ^ v * 2 ) / ( ψ ¨ i + b i 2 σ ^ v * 2 ) , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafq4SdCMbaK aadaqhaaWcbaGaamyAaaqaaiaacQcaaaGccqGH9aqpdaWcgaqaamaa bmaabaGaamOyamaaDaaaleaacaWGPbaabaGaaGOmaaaakiqbeo8aZz aajaWaa0baaSqaaiaadAhaaeaacaGGQaGaaGOmaaaaaOGaayjkaiaa wMcaaaqaamaabmaabaGafqiYdKNbamaadaWgaaWcbaGaamyAaaqaba GccqGHRaWkcaWGIbWaa0baaSqaaiaadMgaaeaacaaIYaaaaOGafq4W dmNbaKaadaqhaaWcbaGaamODaaqaaiaacQcacaaIYaaaaaGccaGLOa GaayzkaaaaaiaacYcaaaa@50AB@ et β ^ * = ( i A z i * z i * T / ( ψ ¨ i + b i 2 σ ^ v * 2 ) ) 1 i A z i * θ ^ i / ( ψ ¨ i + b i 2 σ ^ v * 2 ) . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabCOSdyaaja WaaWbaaSqabeaacaGGQaaaaOGaeyypa0ZaaeWaaeaadaWcgaqaamaa qababaGaaCOEamaaDaaaleaacaWGPbaabaGaaiOkaaaakiaahQhada qhaaWcbaGaamyAaaqaaiaacQcacaWGubaaaaqaaiaadMgacqGHiiIZ caWGbbaabeqdcqGHris5aaGcbaWaaeWaaeaacuaHipqEgaWaamaaBa aaleaacaWGPbaabeaakiabgUcaRiaadkgadaqhaaWcbaGaamyAaaqa aiaaikdaaaGccuaHdpWCgaqcamaaDaaaleaacaWG2baabaGaaiOkai aaikdaaaaakiaawIcacaGLPaaaaaaacaGLOaGaayzkaaWaaWbaaSqa beaacqGHsislcaaIXaaaaOWaaSGbaeaadaaeqaqaaiaahQhadaqhaa WcbaGaamyAaaqaaiaacQcaaaGccuaH4oqCgaqcamaaBaaaleaacaWG PbaabeaaaeaacaWGPbGaeyicI4Saamyqaaqab0GaeyyeIuoaaOqaam aabmaabaGafqiYdKNbamaadaWgaaWcbaGaamyAaaqabaGccqGHRaWk caWGIbWaa0baaSqaaiaadMgaaeaacaaIYaaaaOGafq4WdmNbaKaada qhaaWcbaGaamODaaqaaiaacQcacaaIYaaaaaGccaGLOaGaayzkaaaa aiaac6caaaa@6D17@

Toutes les composantes de θ ^ i EBLUP * , b MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiUdeNbaK aadaqhaaWcbaGaamyAaaqaaiaabweacaqGcbGaaeitaiaabwfacaqG qbWaaWbaaWqabeaacaGGQaaaaSGaaGzaVlaacYcacaaMe8UaamOyaa aaaaa@4274@ sont calculées à l’aide du modèle enrichi obtenu sous (3.5). On peut démontrer que i A A ¯ ω i θ ^ i EBLUP * , b = i A ω i θ ^ i , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaabeaeaacq aHjpWDdaWgaaWcbaGaamyAaaqabaGccuaH4oqCgaqcamaaDaaaleaa caWGPbaabaGaaeyraiaabkeacaqGmbGaaeyvaiaabcfadaahaaadbe qaaiaacQcaaaWccaaMb8UaaiilaiaaysW7caWGIbaaaaqaaiaadMga cqGHiiIZcaWGbbGaeyOkIGSabmyqayaaraaabeqdcqGHris5aOGaey ypa0ZaaabeaeaacqaHjpWDdaWgaaWcbaGaamyAaaqabaGccuaH4oqC gaqcamaaBaaaleaacaWGPbaabeaaaeaacaWGPbGaeyicI4Saamyqaa qab0GaeyyeIuoakiaacYcaaaa@599E@ et, par conséquent, la réconciliation tient.

Les méthodes du rajustement de la différence et du vecteur enrichi sont deux façons de satisfaire la réconciliation. Wang et coll. (2008) ont suggéré d’autres procédures à employer. Plus précisément, ils ont adapté l’estimateur à réconciliation automatique que You et Rao (2002) avaient conçu dans le contexte du modèle au niveau de l’unité pour le modèle au niveau du domaine. You, Rao et Hidiroglou (2013) ont obtenu un estimateur de l’erreur de prédiction quadratique moyenne et de son biais à l’aide d’un modèle mal spécifié.

3.3  Estimation de l’erreur quadratique moyenne

La fiabilité des estimateurs EBLUP est obtenue au moyen de EQM ( θ ^ i EBLUP ) = E ( θ ^ i EBLUP θ i ) 2 . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeyraiaabg facaqGnbWaaeWaaeaacuaH4oqCgaqcamaaDaaaleaacaWGPbaabaGa aeyraiaabkeacaqGmbGaaeyvaiaabcfaaaaakiaawIcacaGLPaaacq GH9aqpcaWGfbWaaeWaaeaacuaH4oqCgaqcamaaDaaaleaacaWGPbaa baGaaeyraiaabkeacaqGmbGaaeyvaiaabcfaaaGccqGHsislcqaH4o qCdaWgaaWcbaGaamyAaaqabaaakiaawIcacaGLPaaadaahaaWcbeqa aiaaikdaaaGccaGGUaaaaa@5095@ Cette espérance prise par rapport au modèle (3.3) pour l’estimateur non réconcilié, et par rapport au modèle (3.5) pour l’estimateur réconcilié.

Les erreurs quadratiques moyennes (EQM) estimées des estimateurs au niveau du domaine sont présentées au tableau 3.1. La forme particulière des termes g MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4zaaaa@36E3@ et les variances estimées se trouvent dans Rao et Molina (2015) ou dans Estevao et coll. (2015). Pour les estimateurs réconciliés, l’EQM estimée de l’approche de rajustement de la différence a recours à des formules de l’EQM non réconciliées. Dans le cas d’une approche vectorielle enrichie, l’EQM est fondée sur l’enrichissement du vecteur z i T MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCOEamaaDa aaleaacaWGPbaabaGaamivaaaaaaa@38EE@ avec ω i ψ ¨ i . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyYdC3aaS baaSqaaiaadMgaaeqaaOGafqiYdKNbamaadaWgaaWcbaGaamyAaaqa baGccaGGUaaaaa@3C96@


Tableau 3.1
Estimations de l’EQM pour les estimateurs au niveau du domaine
Sommaire du tableau
Le tableau montre les résultats de Estimations de l’EQM pour les estimateurs au niveau du domaine. Les données sont présentées selon Estimateur (titres de rangée) et EQM(figurant comme en-tête de colonne).
Estimateur EQM
Fay-Herriot eqm( θ ^ i FH )={ g 0i + g 1i + g 2i +2 g 3i pouriA z i T var( β ^ ) z i +  b i 2 σ ^ v 2 pouri A ¯ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeyzaiaabg hacaqGTbWaaeWaaeaacuaH4oqCgaqcamaaDaaaleaacaWGPbaabaGa aeOraiaabIeaaaaakiaawIcacaGLPaaacqGH9aqpdaGabaqaauaaba qaciaaaeaacaWGNbWaaSbaaSqaaiaaicdacaWGPbaabeaakiabgUca RiaadEgadaWgaaWcbaGaaGymaiaadMgaaeqaaOGaey4kaSIaam4zam aaBaaaleaacaaIYaGaamyAaaqabaGccqGHRaWkcaaIYaGaam4zamaa BaaaleaacaaIZaGaamyAaaqabaaakeaacaqGWbGaae4Baiaabwhaca qGYbGaaGjbVlaaykW7caWGPbGaeyicI4SaamyqaaqaaiaahQhadaqh aaWcbaGaamyAaaqaaiaadsfaaaGccaGG2bGaaiyyaiaackhadaqada qaaiqahk7agaqcaaGaayjkaiaawMcaaiaahQhadaWgaaWcbaGaamyA aaqabaGccqGHRaWkcaqGGaGaamOyamaaDaaaleaacaWGPbaabaGaaG Omaaaakiqbeo8aZzaajaWaa0baaSqaaiaadAhaaeaacaaIYaaaaaGc baGaaeiCaiaab+gacaqG1bGaaeOCaiaaysW7caaMc8UaamyAaiabgI GiolqadgeagaqeaaaaaiaawUhaaaaa@78EB@
ADM eqm( θ ^ i ADM )={ g 0i + g 1i + g 2i +2 g 3i pouriA z i T var( β ^ ) z i +   b i 2 σ ^ v 2 pouri A ¯ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeyzaiaabg hacaqGTbWaaeWaaeaacuaH4oqCgaqcamaaDaaaleaacaWGPbaabaGa aeyqaiaabseacaqGnbaaaaGccaGLOaGaayzkaaGaeyypa0Zaaiqaae aafaqaaeGacaaabaGaam4zamaaBaaaleaacaaIWaGaamyAaaqabaGc cqGHRaWkcaWGNbWaaSbaaSqaaiaaigdacaWGPbaabeaakiabgUcaRi aadEgadaWgaaWcbaGaaGOmaiaadMgaaeqaaOGaey4kaSIaaGOmaiaa dEgadaWgaaWcbaGaaG4maiaadMgaaeqaaaGcbaGaaeiCaiaab+gaca qG1bGaaeOCaiaaysW7caaMc8UaamyAaiabgIGiolaadgeaaeaacaWH 6bWaa0baaSqaaiaadMgaaeaacaWGubaaaOGaaiODaiaacggacaGGYb WaaeWaaeaaceWHYoGbaKaaaiaawIcacaGLPaaacaWH6bWaaSbaaSqa aiaadMgaaeqaaOGaey4kaSIaaeiiaiaabccacaWGIbWaa0baaSqaai aadMgaaeaacaaIYaaaaOGafq4WdmNbaKaadaqhaaWcbaGaamODaaqa aiaaikdaaaaakeaacaqGWbGaae4BaiaabwhacaqGYbGaaGjbVlaayk W7caWGPbGaeyicI4SabmyqayaaraaaaaGaay5Eaaaaaa@7A55@
REML eqm( θ ^ i REML )={ g 1i + g 2i +2 g 3i pouriA z i T var( β ^ ) z i +  b i 2 σ ^ v 2 pouri A ¯ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeyzaiaabg hacaqGTbWaaeWaaeaacuaH4oqCgaqcamaaDaaaleaacaWGPbaabaGa aeOuaiaabweacaqGnbGaaeitaaaaaOGaayjkaiaawMcaaiabg2da9m aaceaabaqbaeaabiGaaaqaaiaadEgadaWgaaWcbaGaaGymaiaadMga aeqaaOGaey4kaSIaam4zamaaBaaaleaacaaIYaGaamyAaaqabaGccq GHRaWkcaaIYaGaam4zamaaBaaaleaacaaIZaGaamyAaaqabaaakeaa caqGWbGaae4BaiaabwhacaqGYbGaaGjbVlaaykW7caWGPbGaeyicI4 SaamyqaaqaaiaahQhadaqhaaWcbaGaamyAaaqaaiaadsfaaaGccaqG 2bGaaeyyaiaabkhadaqadaqaaiqahk7agaqcaaGaayjkaiaawMcaai aahQhadaWgaaWcbaGaamyAaaqabaGccqGHRaWkcaqGGaGaamOyamaa DaaaleaacaWGPbaabaGaaGOmaaaakiqbeo8aZzaajaWaa0baaSqaai aadAhaaeaacaaIYaaaaaGcbaGaaeiCaiaab+gacaqG1bGaaeOCaiaa ysW7caaMc8UaamyAaiabgIGiolqadgeagaqeaaaaaiaawUhaaaaa@76E4@
WF eqm( θ ^ i WF  )={ g 1i + g 2i +2 g 3i + g 4i pouriA z i T var( β ^ ) z i + b i 2 σ ^ v 2 pouri A ¯ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeyzaiaabg hacaqGTbWaaeWaaeaacuaH4oqCgaqcamaaDaaaleaacaWGPbaabaGa ae4vaiaabAeacaqGGaaaaaGccaGLOaGaayzkaaGaeyypa0Zaaiqaae aafaqaaeGacaaabaGaam4zamaaBaaaleaacaaIXaGaamyAaaqabaGc cqGHRaWkcaWGNbWaaSbaaSqaaiaaikdacaWGPbaabeaakiabgUcaRi aaikdacaWGNbWaaSbaaSqaaiaaiodacaWGPbaabeaakiabgUcaRiaa dEgadaWgaaWcbaGaaGinaiaadMgaaeqaaaGcbaGaaeiCaiaab+gaca qG1bGaaeOCaiaaysW7caaMc8UaamyAaiabgIGiolaadgeaaeaacaWH 6bWaa0baaSqaaiaadMgaaeaacaWGubaaaOGaaeODaiaabggacaqGYb WaaeWaaeaaceWHYoGbaKaaaiaawIcacaGLPaaacaWH6bWaaSbaaSqa aiaadMgaaeqaaOGaey4kaSIaamOyamaaDaaaleaacaWGPbaabaGaaG Omaaaakiqbeo8aZzaajaWaa0baaSqaaiaadAhaaeaacaaIYaaaaaGc baGaaeiCaiaab+gacaqG1bGaaeOCaiaaysW7caaMc8UaamyAaiabgI GiolqadgeagaqeaaaaaiaawUhaaaaa@78FB@

Les divers termes g MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4zaaaa@36E3@ dans le tableau 3.1 peuvent être interprétés comme suit. Le   g 0 i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeiiaiaadE gadaWgaaWcbaGaaGimaiaadMgaaeqaaaaa@395A@ est un terme de correction du biais pour les méthodes FH et ADM. Le terme g 1 i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4zamaaBa aaleaacaaIXaGaamyAaaqabaaaaa@38B8@ obtenu à l’aide de g 1 i = γ ^ i ψ ¨ i , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4zamaaBa aaleaacaaIXaGaamyAaaqabaGccqGH9aqpcuaHZoWzgaqcamaaBaaa leaacaWGPbaabeaakiqbeI8a5zaadaWaaSbaaSqaaiaadMgaaeqaaO Gaaiilaaaa@404F@ représente la majeure partie de l’EQM si le nombre de domaines est élevé. Le terme g 2 i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4zamaaBa aaleaacaaIYaGaamyAaaqabaaaaa@38B9@ représente l’estimation de β , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCOSdiaacY caaaa@37E5@ et 2 g 3 i , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOmaiaadE gadaWgaaWcbaGaaG4maiaadMgaaeqaaOGaaiilaaaa@3A30@ l’estimation de σ v 2 . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aa0 baaSqaaiaadAhaaeaacaaIYaaaaOGaaiOlaaaa@3A5A@ Le terme g 4 i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4zamaaBa aaleaacaaI0aGaamyAaaqabaaaaa@38BB@ dans la procédure WF indique que la valeur estimée de ψ i , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiYdK3aaS baaSqaaiaadMgaaeqaaOGaaiilaaaa@3999@ ψ ^ i , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiYdKNbaK aadaWgaaWcbaGaamyAaaqabaGccaGGSaaaaa@39A9@ a été utilisée. La variance estimée de β ^ , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabCOSdyaaja Gaaiilaaaa@37F5@ obtenue par var ( β ^ ) = ( i A z i z i T ψ ¨ i + b i 2 σ ^ v 2 ) 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbiqaaaGecaqG2b Gaaeyyaiaabkhadaqadaqaaiqahk7agaqcaaGaayjkaiaawMcaaiab g2da9maabmaabaWaaabeaeaadaWcbaWcbaGaaCOEamaaBaaameaaca WGPbaabeaaliaahQhadaqhaaadbaGaamyAaaqaaiaadsfaaaaaleaa cuaHipqEgaWaamaaBaaameaacaWGPbaabeaaliabgUcaRiaadkgada qhaaadbaGaamyAaaqaaiaaikdaaaWccuaHdpWCgaqcamaaDaaameaa caWG2baabaGaaGOmaaaaaaaaleaacaWGPbGaeyicI4Saamyqaaqab0 GaeyyeIuoaaOGaayjkaiaawMcaamaaCaaaleqabaGaeyOeI0IaaGym aaaaaaa@54E8@ dépend de la procédure particulière employée pour estimer σ v 2 . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aa0 baaSqaaiaadAhaaeaacaaIYaaaaOGaaiOlaaaa@3A5A@


Date de modification :