Estimation sur petits domaines portant sur des chiffres pondérés d’enquête dans un modèle spatial à niveau agrégé

Section 3. Estimation sur petits domaines avec le MLGM au niveau du domaine

3.1  Effets aléatoires de domaine sans corrélation spatiale

Nous supposons qu’un échantillonnage probabiliste permet de tirer un échantillon s MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4Caaaa@36EF@ de taille n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaaaa@36EA@ d’une population finie U MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyvaaaa@36D1@ de taille N MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOtaaaa@36CA@ consistant en D MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiraaaa@36C0@ domaines disjoints U i ( i = 1 , , D ) . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyvamaaBa aaleaacaWGPbaabeaakmaabmaabaGaamyAaiabg2da9iaaigdacaGG SaGaaGjbVlablAciljaacYcacaaMe8UaamiraaGaayjkaiaawMcaai aac6caaaa@4344@ Comme c’est la règle, nous parlerons ici de petits domaines ou seulement de domaines. Nous posons également qu’il y a un nombre connu N i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOtamaaBa aaleaacaWGPbaabeaaaaa@37E4@ d’unités de population dans le petit domaine i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAaaaa@36E5@ et que n i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBamaaBa aaleaacaWGPbaabeaaaaa@3804@ de ces unités sont échantillonnées. Le nombre total d’unités de la population est N = i = 1 D N i , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOtaiabg2 da9maaqadabaGaamOtamaaBaaaleaacaWGPbaabeaaaeaacaWGPbGa eyypa0JaaGymaaqaaiaadseaa0GaeyyeIuoakiaacYcaaaa@3FE6@ et la taille totale d’échantillon correspondante, n = i = 1 D n i . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaiabg2 da9maaqadabaGaamOBamaaBaaaleaacaWGPbaabeaaaeaacaWGPbGa eyypa0JaaGymaaqaaiaadseaa0GaeyyeIuoakiaac6caaaa@4028@ Soit s MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4Caaaa@36EF@ la collection d’unités de l’échantillon et s i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4CamaaBa aaleaacaWGPbaabeaaaaa@3809@ le sous-ensemble tiré du petit domaine i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAaaaa@36E5@ ( | s i | = n i ) , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaaca GG8bGaaGPaVlaadohadaWgaaWcbaGaamyAaaqabaGccaaMc8UaaiiF aiaaysW7cqGH9aqpcaaMe8UaamOBamaaBaaaleaacaWGPbaabeaaaO GaayjkaiaawMcaaiaacYcaaaa@4599@ nous employons des expressions comme j i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOAaiabgI GiolaadMgaaaa@3958@ et j s MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOAaiabgI Giolaadohaaaa@3962@ pour les unités composant le petit domaine i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAaaaa@36E5@ et l’échantillon s MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4Caaaa@36EF@ respectivement. De même, r i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCamaaBa aaleaacaWGPbaabeaaaaa@3808@ désigne l’ensemble d’unités du petit domaine i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAaaaa@36E5@ qui ne sont pas dans l’échantillon avec | r i | = N i n i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiiFaiaayk W7caWGYbWaaSbaaSqaaiaadMgaaeqaaOGaaGPaVlaacYhacaaMe8Ua eyypa0JaaGjbVlaad6eadaWgaaWcbaGaamyAaaqabaGccqGHsislca WGUbWaaSbaaSqaaiaadMgaaeqaaaaa@4639@ et U i = s i r i . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyvamaaBa aaleaacaWGPbaabeaakiabg2da9iaadohadaWgaaWcbaGaamyAaaqa baGccqGHQicYcaWGYbWaaSbaaSqaaiaadMgaaeqaaOGaaiOlaaaa@3F84@ Soit y i j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEamaaBa aaleaacaWGPbGaamOAaaqabaaaaa@38FE@ la valeur de la variable d’intérêt pour l’unité j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOAaaaa@36E6@ ( j = 1 , , N i ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaaca WGQbGaeyypa0JaaGymaiaacYcacaaMe8UaeSOjGSKaaiilaiaaysW7 caWGobWaaSbaaSqaaiaadMgaaeqaaaGccaGLOaGaayzkaaaaaa@41C3@ dans le domaine i . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAaiaac6 caaaa@3797@ La variable d’intérêt aux valeurs y i i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEamaaBa aaleaacaWGPbGaamyAaaqabaaaaa@38FD@ est binaire ( y i j = 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadM hadaWgaaWcbaGaamyAaiaadQgaaeqaaOGaeyypa0JaaGymaaaa@3B75@ si le ménage j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOAaaaa@36E6@ dans le domaine i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAaaaa@36E5@ est pauvre et 0 dans les autres cas). Le but est d’estimer les chiffres de population de petits domaines y i = j U i y i j , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEamaaBa aaleaacaWGPbaabeaakiabg2da9maaqababaGaamyEamaaBaaaleaa caWGPbGaamOAaaqabaaabaGaamOAaiabgIGiolaadwfadaWgaaadba GaamyAaaqabaaaleqaniabggHiLdGccaGGSaaaaa@432B@ ou, comme équivalent, la proportion de petits domaines P i = N i 1 j U i y i j , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiuamaaBa aaleaacaWGPbaabeaakiabg2da9iaad6eadaqhaaWcbaGaamyAaaqa aiabgkHiTiaaigdaaaGcdaaeqaqaaiaadMhadaWgaaWcbaGaamyAai aadQgaaeqaaaqaaiaadQgacqGHiiIZcaWGvbWaaSbaaWqaaiaadMga aeqaaaWcbeqdcqGHris5aOGaaiilaaaa@46A2@ dans le domaine i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAaaaa@36E5@ ( i = 1 , , D ) . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaaca WGPbGaeyypa0JaaGymaiaacYcacaaMe8UaeSOjGSKaaiilaiaaysW7 caWGebaacaGLOaGaayzkaaGaaiOlaaaa@4146@ L’estimateur direct type d’enquête (DIR dans la suite du texte) pour P i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiuamaaBa aaleaacaWGPbaabeaaaaa@37E6@ est p i w = j s i w ˜ i j y i j , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCamaaBa aaleaacaWGPbGaam4DaaqabaGccqGH9aqpdaaeqaqaaiqadEhagaac amaaBaaaleaacaWGPbGaamOAaaqabaGccaWG5bWaaSbaaSqaaiaadM gacaWGQbaabeaaaeaacaWGQbGaeyicI4Saam4CamaaBaaameaacaWG PbaabeaaaSqab0GaeyyeIuoakiaacYcaaaa@475A@ w ˜ i j = w i j / j s i w i j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabm4Dayaaia WaaSbaaSqaaiaadMgacaWGQbaabeaakiabg2da9maalyaabaGaam4D amaaBaaaleaacaWGPbGaamOAaaqabaaakeaadaaeqaqaaiaadEhada WgaaWcbaGaamyAaiaadQgaaeqaaaqaaiaadQgacqGHiiIZcaWGZbWa aSbaaWqaaiaadMgaaeqaaaWcbeqdcqGHris5aaaaaaa@46AE@ est le poids d’échantillon normalisé pour l’unité j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOAaaaa@36E6@ dans le domaine i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAaaaa@36E5@ avec j s i w ˜ i j = 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaabeaeaace WG3bGbaGaadaWgaaWcbaGaamyAaiaadQgaaeqaaaqaaiaadQgacqGH iiIZcaWGZbWaaSbaaWqaaiaadMgaaeqaaaWcbeqdcqGHris5aOGaey ypa0JaaGymaaaa@413F@ et où w i j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4DamaaBa aaleaacaWGPbGaamOAaaqabaaaaa@38FC@ est le poids d’enquête pour l’unité j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOAaaaa@36E6@ dans le domaine i . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAaiaac6 caaaa@3797@ On approche la variance de plan de sondage estimée de DIR par v ( p i w ) j s i w ˜ i j ( w ˜ i j 1 ) ( y i j p i w ) 2 . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODamaabm aabaGaamiCamaaBaaaleaacaWGPbGaam4DaaqabaaakiaawIcacaGL PaaacqGHijYUdaaeqaqaaiqadEhagaacamaaBaaaleaacaWGPbGaam OAaaqabaGcdaqadaqaaiqadEhagaacamaaBaaaleaacaWGPbGaamOA aaqabaGccqGHsislcaaIXaaacaGLOaGaayzkaaWaaeWaaeaacaWG5b WaaSbaaSqaaiaadMgacaWGQbaabeaakiabgkHiTiaadchadaWgaaWc baGaamyAaiaadEhaaeqaaaGccaGLOaGaayzkaaaaleaacaWGQbGaey icI4Saam4CamaaBaaameaacaWGPbaabeaaaSqab0GaeyyeIuoakmaa CaaaleqabaGaaGOmaaaakiaac6caaaa@576D@ On emploie cette formule pour l’estimateur de variance de DIR d’après Särndal, Swensson et Wretman (1992; voir pages 43, 185 et 391), et après les simplifications w i j = a i j 1 , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4DamaaBa aaleaacaWGPbGaamOAaaqabaGccqGH9aqpcaWGHbWaaSbaaSqaaiaa dMgacaWGQbaabeaakmaaCaaaleqabaGaeyOeI0IaaGymaaaakiaacY caaaa@3F94@ a i j , i j = a i j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaBa aaleaacaWGPbGaamOAaiaaygW7caGGSaGaaGPaVlaadMgacaWGQbaa beaakiabg2da9iaadggadaWgaaWcbaGaamyAaiaadQgaaeqaaaaa@4287@ et a i j , i k = a i j a i k , j k , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaBa aaleaacaWGPbGaamOAaiaaygW7caGGSaGaaGPaVlaadMgacaWGRbaa beaakiabg2da9iaadggadaWgaaWcbaGaamyAaiaadQgaaeqaaOGaam yyamaaBaaaleaacaWGPbGaam4AaaqabaGccaGGSaGaaGjbVlaadQga cqGHGjsUcaWGRbGaaiilaaaa@4C1F@ a i j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaBa aaleaacaWGPbGaamOAaaqabaaaaa@38E6@ est la probabilité d’inclusion de premier ordre de l’unité j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOAaaaa@36E6@ dans le domaine i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAaaaa@36E5@ et où a i j , i k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaBa aaleaacaWGPbGaamOAaiaaygW7caGGSaGaaGPaVlaadMgacaWGRbaa beaaaaa@3E89@ est la probabilité d’inclusion de second ordre des unités j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOAaaaa@36E6@ et k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4Aaaaa@36E7@ dans le domaine i . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAaiaac6 caaaa@3797@ Dans un échantillonnage aléatoire simple (EAS), w i j = N i n i 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4DamaaBa aaleaacaWGPbGaamOAaaqabaGccqGH9aqpcaWGobWaaSbaaSqaaiaa dMgaaeqaaOGaamOBamaaDaaaleaacaWGPbaabaGaeyOeI0IaaGymaa aaaaa@3FB9@ et DIR est alors p i = n i 1 y s i , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCamaaBa aaleaacaWGPbaabeaakiabg2da9iaad6gadaqhaaWcbaGaamyAaaqa aiabgkHiTiaaigdaaaGccaWG5bWaaSbaaSqaaiaadohacaWGPbaabe aakiaacYcaaaa@40A0@ avec une variance estimée v ( p i ) n i 1 p i ( 1 p i ) , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODamaabm aabaGaamiCamaaBaaaleaacaWGPbaabeaaaOGaayjkaiaawMcaaiab gIKi7kaad6gadaqhaaWcbaGaamyAaaqaaiabgkHiTiaaigdaaaGcca WGWbWaaSbaaSqaaiaadMgaaeqaaOWaaeWaaeaacaaIXaGaeyOeI0Ia amiCamaaBaaaleaacaWGPbaabeaaaOGaayjkaiaawMcaaiaacYcaaa a@4818@ y s i = j s i y i j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEamaaBa aaleaacaWGZbGaamyAaaqabaGccqGH9aqpdaaeqaqaaiaadMhadaWg aaWcbaGaamyAaiaadQgaaeqaaaqaaiaadQgacqGHiiIZcaWGZbWaaS baaWqaaiaadMgaaeqaaaWcbeqdcqGHris5aaaa@4387@ désigne le chiffre de dénombrement d’échantillon dans le domaine i . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAaiaac6 caaaa@3797@

Si le plan de sondage est informatif, la version EAS de DIR définie au paragraphe précédent peut être entachée d’un biais. Dans le cas de l’EDCM 2011-2012, nous avons donc calculé des effets de plan de sondage par district définis comme le rapport entre la variance des estimations pondérées (pour un plan donné d’échantillonnage) et la variance des estimations non pondérées (pour un échantillonnage aléatoire simple). On pouvait constater que ces effets étaient supérieurs à l’unité dans tous les districts sauf trois, les valeurs relevées variant de 1,17 à 8,44 pour une moyenne de 2,71. On est nettement porté à penser que le plan d’échantillonnage de l’EDCM de 2011-2012 est informatif. Ajoutons que l’estimateur direct d’enquête DIR repose sur des données d’échantillon par domaine et peut donc se révéler fort imprécis si la taille d’échantillon par domaine est petite, voire impossible à calculer si cette même taille est nulle. Pour résoudre ce problème, nous pouvons néanmoins employer des méthodes d’EPD par modèle qui « empruntent de la puissance » grâce à un modèle statistique commun pour tous les petits domaines en question (voir Rao et Molina (2015)).

Posons maintenant que les données disponibles consistent en agrégats d’échantillon y s i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEamaaBa aaleaacaWGZbGaamyAaaqabaaaaa@3907@ (dénombrements d’échantillon des ménages pauvres) avec les valeurs de covariables de contexte par domaine. Ainsi, nous obtenons le chiffre de dénombrement y s i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEamaaBa aaleaacaWGZbGaamyAaaqabaaaaa@3907@ pour le domaine i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAaaaa@36E4@ avec un vecteur k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4Aaaaa@36E6@ de covariables x i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbeGaa8hEam aaBaaaleaacaWGPbaabeaaaaa@3814@ par domaine venant de sources d’information secondaires (recensement ou registres administratifs, par exemple). Si nous ne tenons pas compte du plan d’échantillonnage, nous pouvons supposer que le dénombrement d’échantillon y s i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEamaaBa aaleaacaWGZbGaamyAaaqabaaaaa@3907@ suit une distribution binominale avec les paramètres n i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBamaaBa aaleaacaWGPbaabeaaaaa@3804@ et π i , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiWda3aaS baaSqaaiaadMgaaeqaaOGaaiilaaaa@3988@ soit y s i Bin ( n i , π i ) , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEamaaBa aaleaacaWGZbGaamyAaaqabaGccaaMc8EeeuuDJXwAKbsr4rNCHbac faGae8hpIOJaaGPaVlaaysW7caqGcbGaaeyAaiaab6gadaqadaqaai aad6gadaWgaaWcbaGaamyAaaqabaGccaGGSaGaaGjbVlabec8aWnaa BaaaleaacaWGPbaabeaaaOGaayjkaiaawMcaaiaacYcaaaa@4F7D@ π i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiWda3aaS baaSqaaiaadMgaaeqaaaaa@38CE@ est la probabilité d’occurrence pour une unité de population dans le domaine i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAaaaa@36E5@ ou la probabilité de fréquence dans ce même domaine. D’après Saei et Chambers (2003), Johnson et coll. (2010) et Chandra et coll. (2011), ce modèle couplant la probabilité π i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiWda3aaS baaSqaaiaadMgaaeqaaaaa@38CE@ et les covariables x i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbeGaa8hEam aaBaaaleaacaWGPbaabeaaaaa@3814@ est le modèle linéaire mixte logistique de la forme

logit ( π i ) = ln { π i ( 1 π i ) 1 } = η i = x i T β + u i , ( 3.1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeiBaiaab+ gacaqGNbGaaeyAaiaabshadaqadaqaaiabec8aWnaaBaaaleaacaWG PbaabeaaaOGaayjkaiaawMcaaiabg2da9iGacYgacaGGUbWaaiWaae aacqaHapaCdaWgaaWcbaGaamyAaaqabaGcdaqadaqaaiaaigdacqGH sislcqaHapaCdaWgaaWcbaGaamyAaaqabaaakiaawIcacaGLPaaada ahaaWcbeqaaiabgkHiTiaaigdaaaaakiaawUhacaGL9baacqGH9aqp iiGacqWF3oaAdaWgaaWcbaGaamyAaaqabaGccqGH9aqpcaWH4bWaa0 baaSqaaiaadMgaaeaacaWGubaaaOGaaCOSdiabgUcaRiaadwhadaWg aaWcbaGaamyAaaqabaGccaGGSaGaaGzbVlaaywW7caaMf8UaaGzbVl aaywW7caGGOaGaaG4maiaac6cacaaIXaGaaiykaaaa@6714@

avec π i = exp ( x i T β + u i ) { 1 + exp ( x i T β + u i ) } 1 = expit ( x i T β + u i ) . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiWda3aaS baaSqaaiaadMgaaeqaaOGaeyypa0JaciyzaiaacIhacaGGWbWaaeWa aeaacaWH4bWaa0baaSqaaiaadMgaaeaacaWGubaaaOGaaCOSdiabgU caRiaadwhadaWgaaWcbaGaamyAaaqabaaakiaawIcacaGLPaaadaGa daqaaiaaigdacqGHRaWkciGGLbGaaiiEaiaacchadaqadaqaaiaahI hadaqhaaWcbaGaamyAaaqaaiaadsfaaaGccaWHYoGaey4kaSIaamyD amaaBaaaleaacaWGPbaabeaaaOGaayjkaiaawMcaaaGaay5Eaiaaw2 haamaaCaaaleqabaGaeyOeI0IaaGymaaaakiabg2da9iGacwgacaGG 4bGaaiiCaiaacMgacaGG0bWaaeWaaeaacaWH4bWaa0baaSqaaiaadM gaaeaacaWGubaaaOGaaCOSdiabgUcaRiaadwhadaWgaaWcbaGaamyA aaqabaaakiaawIcacaGLPaaacaGGUaaaaa@660A@ Là, β MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCOSdaaa@3734@ est le vecteur k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4Aaaaa@36E6@ de coefficients de régression, ce qu’on appelle souvent le vecteur des effets fixes, et u i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyDamaaBa aaleaacaWGPbaabeaaaaa@380B@ est l’effet aléatoire par domaine avec u i N ( 0 , σ u 2 ) . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyDamaaBa aaleaacaWGPbaabeaarqqr1ngBPrgifHhDYfgaiuaakiab=XJi6iaa d6eadaqadaqaaiaaicdacaGGSaGaeq4Wdm3aa0baaSqaaiaadwhaae aacaaIYaaaaaGccaGLOaGaayzkaaGaaiOlaaaa@45F6@ Le modèle (3.1) est un cas d’espèce du modèle linéaire généralisé mixte (MLGM) avec une fonction de lien logits (Breslow et Clayton, 1993). Nous pouvons observer que les paramètres β MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCOSdaaa@3735@ et σ u 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aa0 baaSqaaiaadwhaaeaacaaIYaaaaaaa@399D@ sont les mêmes pour chaque domaine, c’est-à-dire qu’ils peuvent s’estimer à l’aide des données de tous les petits domaines. C’est ce que l’on fait habituellement en « empilant » les modèles D au niveau du domaine en (3.1) pour produire au niveau de la population un modèle de la forme

η = X β + Z u ( 3.2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaC4Tdiabg2 da9iaahIfacaWHYoGaey4kaSIaaCOwaiaahwhacaaMf8UaaGzbVlaa ywW7caaMf8UaaGzbVlaacIcacaaIZaGaaiOlaiaaikdacaGGPaaaaa@486B@

avec π = ( π 1 , , π D ) T = expit ( η ) , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCiWdiabg2 da9maabmaabaGaeqiWda3aaSbaaSqaaiaaigdaaeqaaOGaaiilaiaa ysW7cqWIMaYscaGGSaGaaGjbVlabec8aWnaaBaaaleaacaWGebaabe aaaOGaayjkaiaawMcaamaaCaaaleqabaGaamivaaaakiabg2da9iaa bwgacaqG4bGaaeiCaiaabMgacaqG0bWaaeWaaeaacaWH3oaacaGLOa GaayzkaaGaaiilaaaa@4F23@ η = ( η 1 , , η D ) T , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaC4Tdiabg2 da9maabmaabaGaeq4TdG2aaSbaaSqaaiaaigdaaeqaaOGaaiilaiaa ysW7cqWIMaYscaGGSaGaaGjbVlabeE7aOnaaBaaaleaacaWGebaabe aaaOGaayjkaiaawMcaamaaCaaaleqabaGaamivaaaakiaacYcaaaa@466D@ X = ( x 1 T , , x D T ) T MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCiwaiabg2 da9maabmaabaGaaCiEamaaDaaaleaacaaIXaaabaGaamivaaaakiaa cYcacaaMe8UaeSOjGSKaaiilaiaaysW7caWH4bWaa0baaSqaaiaads eaaeaacaWGubaaaaGccaGLOaGaayzkaaWaaWbaaSqabeaacaWGubaa aaaa@45AF@ est la matrice D × k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiraiabgE na0kaadUgaaaa@39C7@ des covariables, où Z = ( z 1 , , z D ) T MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCOwaiabg2 da9maabmaabaGaaCOEamaaBaaaleaacaaIXaaabeaakiaacYcacaaM e8UaeSOjGSKaaiilaiaaysW7caWH6bWaaSbaaSqaaiaadseaaeqaaa GccaGLOaGaayzkaaWaaWbaaSqabeaacaWGubaaaaaa@4401@ est une matrice de covariables connues de dimension D × D MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiraiabgE na0kaadseaaaa@39A0@ caractérisant les différences entre les petits domaines, où z i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCOEamaaBa aaleaacaWGPbaabeaaaaa@3814@ est le vecteur D MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiraaaa@36BF@ ( 0 , , 1 , , 0 ) T MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaaca aIWaGaaiilaiaaysW7cqWIMaYscaGGSaGaaGjbVlaaigdacaGGSaGa aGjbVlablAciljaacYcacaaMe8UaaGimaaGaayjkaiaawMcaamaaCa aaleqabaGaamivaaaaaaa@45ED@ avec le 1 dans la i e MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAamaaCa aaleqabaGaaeyzaaaaaaa@37F9@ position et où u = ( u 1 , , u D ) T MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCyDaiabg2 da9iabgwMiZoaabmaabaGaamyDamaaBaaaleaacaaIXaaabeaakiaa cYcacaaMe8UaeSOjGSKaaiilaiaaysW7caWG1bWaaSbaaSqaaiaads eaaeqaaaGccaGLOaGaayzkaaWaaWbaaSqabeaacaWGubaaaaaa@45D0@ est le vecteur D MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiraaaa@36BF@ des effets aléatoires de domaine avec u N ( 0 , Ω ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCyDaebbfv 3ySLgzGueE0jxyaGqbaiab=XJi6iaad6eadaqadaqaaiaahcdacaGG SaGaaGjbVlaahM6aaiaawIcacaGLPaaaaaa@4335@ Ω = Ω ( σ u 2 ) = σ u 2 I D MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCyQdiabg2 da9iaahM6adaqadaqaaiabeo8aZnaaDaaaleaacaWG1baabaGaaGOm aaaaaOGaayjkaiaawMcaaiabg2da9iabeo8aZnaaDaaaleaacaWG1b aabaGaaGOmaaaakiaahMeadaWgaaWcbaGaamiraaqabaaaaa@451D@ et I D MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCysamaaBa aaleaacaWGebaabeaaaaa@37BE@ est une matrice diagonale D × D MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiraiabgE na0kaadseaaaa@39A0@ .

Le traitement de quasi-vraisemblance pénalisée (QVP) est une procédure d’estimation largement utilisée dans l’ajustement d’un MLGM. Il s’agit d’établir une approximation linéaire de la variable de réponse non normale, puis de poser que cette variable dépendante linéarisée est approximativement normale. La méthode QVP sert largement à l’estimation sur petits domaines, puisqu’elle est bien plus facile à appliquer dans la pratique que le meilleur prédicteur empirique. Précisons que nous employons une méthode hybride avec le traitement QVP pour l’estimation des paramètres β MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCOSdaaa@3735@ et u MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCyDaaaa@36F5@ dans le MLGM (3.2) et le traitement de maximum de vraisemblance restreint pour l’estimation du paramètre de variance Ω ( σ u 2 ) . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCyQdmaabm aabaGaeq4Wdm3aa0baaSqaaiaadwhaaeaacaaIYaaaaaGccaGLOaGa ayzkaaGaaiOlaaaa@3D17@ Voir à ce sujet Breslow et Clayton (1993), Saei et Chambers (2003) et Manteiga, Lombardia, Molina, Morales et Santamarìa (2007). Il est connu que, dans certains cas, le traitement QVP peut mener à des estimateurs incohérents, mais de récentes applications empiriques de la procédure QVP (Manteiga et coll., 2007), font voir que la méthode fonctionne bien dans la pratique.

Dans le modèle (3.2), les valeurs attendues de y s i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEamaaBa aaleaacaWGZbGaamyAaaqabaaaaa@3907@ et y r i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEamaaBa aaleaacaWGYbGaamyAaaqabaaaaa@3906@ étant donné u i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyDamaaBa aaleaacaWGPbaabeaaaaa@380B@ viennent respectivement de μ s i = E ( y s i | u i ) = n i expit ( x i T β + z i T u ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiVd02aaS baaSqaaiaadohacaWGPbaabeaakiabg2da9iaadweadaqadaqaamaa eiaabaGaamyEamaaBaaaleaacaWGZbGaamyAaaqabaGccaaMc8oaca GLiWoacaaMc8UaamyDamaaBaaaleaacaWGPbaabeaaaOGaayjkaiaa wMcaaiabg2da9iaad6gadaWgaaWcbaGaamyAaaqabaGccaaMe8Uaci yzaiaacIhacaGGWbGaaiyAaiaacshadaqadaqaaiaahIhadaqhaaWc baGaamyAaaqaaiaadsfaaaGccaWHYoGaey4kaSIaaCOEamaaDaaale aacaWGPbaabaGaamivaaaakiaahwhaaiaawIcacaGLPaaaaaa@5B17@ et μ r i = E ( y r i | u i ) = ( N i n i ) expit ( x i T β + z i T u ) . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiVd02aaS baaSqaaiaadkhacaWGPbaabeaakiabg2da9iaadweadaqadaqaamaa eiaabaGaamyEamaaBaaaleaacaWGYbGaamyAaaqabaGccaaMe8oaca GLiWoacaaMe8UaamyDamaaBaaaleaacaWGPbaabeaaaOGaayjkaiaa wMcaaiabg2da9maabmaabaGaamOtamaaBaaaleaacaWGPbaabeaaki abgkHiTiaad6gadaWgaaWcbaGaamyAaaqabaaakiaawIcacaGLPaaa ciGGLbGaaiiEaiaacchacaGGPbGaaiiDamaabmaabaGaaCiEamaaDa aaleaacaWGPbaabaGaamivaaaakiaahk7acqGHRaWkcaWH6bWaa0ba aSqaaiaadMgaaeaacaWGubaaaOGaaCyDaaGaayjkaiaawMcaaiaac6 caaaa@5EAB@ Avec (3.2), un prédicteur plug-in empirique (PE) du chiffre de population y i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEamaaBa aaleaacaWGPbaabeaaaaa@380F@ dans le domaine i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAaaaa@36E5@ est

y ^ i PE = y s i + μ ^ r i = y s i + ( N i n i ) π ^ i PE , ( 3.3 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmyEayaaja Waa0baaSqaaiaadMgaaeaacaqGqbGaaeyraaaakiabg2da9iaadMha daWgaaWcbaGaam4CaiaadMgaaeqaaOGaey4kaSIafqiVd0MbaKaada WgaaWcbaGaamOCaiaadMgaaeqaaOGaeyypa0JaamyEamaaBaaaleaa caWGZbGaamyAaaqabaGccqGHRaWkdaqadaqaaiaad6eadaWgaaWcba GaamyAaaqabaGccqGHsislcaWGUbWaaSbaaSqaaiaadMgaaeqaaaGc caGLOaGaayzkaaGaaGjbVlqbec8aWzaajaWaa0baaSqaaiaadMgaae aacaqGqbGaaeyraaaakiaacYcacaaMf8UaaGzbVlaaywW7caaMf8Ua aGzbVlaacIcacaaIZaGaaiOlaiaaiodacaGGPaaaaa@6042@

π ^ i PE = expit ( x i T β ^ + z i T u ^ ) . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiWdaNbaK aadaqhaaWcbaGaamyAaaqaaiaabcfacaqGfbaaaOGaeyypa0Jaciyz aiaacIhacaGGWbGaaiyAaiaacshadaqadaqaaiaahIhadaqhaaWcba GaamyAaaqaaiaadsfaaaGcceWHYoGbaKaacqGHRaWkcaWH6bWaa0ba aSqaaiaadMgaaeaacaWGubaaaOGabCyDayaajaaacaGLOaGaayzkaa GaaiOlaaaa@4BC3@ On obtient une estimation de la proportion correspondante dans le domaine  i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAaaaa@36E5@ comme P ^ i PE = N i 1 y ^ i PE . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmiuayaaja Waa0baaSqaaiaadMgaaeaacaqGqbGaaeyraaaakiabg2da9iaad6ea daqhaaWcbaGaamyAaaqaaiabgkHiTiaaigdaaaGcceWG5bGbaKaada qhaaWcbaGaamyAaaqaaiaabcfacaqGfbaaaOGaaiOlaaaa@42C2@ Pour le domaine i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAaaaa@36E5@ non échantillonné avec le vecteur lié de covariables x i , out , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbeGaa8hEam aaBaaaleaacaWGPbGaaGjcVlaacYcacaaMc8Uaae4BaiaabwhacaqG 0baabeaakiaacYcaaaa@3F7B@ l’estimateur synthétique de y i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEamaaBa aaleaacaWGPbaabeaaaaa@380F@ dans (3.2) est y ^ i SYN = N i π ^ i SYN , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmyEayaaja Waa0baaSqaaiaadMgaaeaacaqGtbGaaeywaiaab6eaaaGccqGH9aqp caWGobWaaSbaaSqaaiaadMgaaeqaaOGafqiWdaNbaKaadaqhaaWcba GaamyAaaqaaiaabofacaqGzbGaaeOtaaaakiaacYcaaaa@43CF@ avec π ^ i SYN = expit ( x i , out T β ^ ) . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiWdaNbaK aadaqhaaWcbaGaamyAaaqaaiaabofacaqGzbGaaeOtaaaakiabg2da 9iGacwgacaGG4bGaaiiCaiaacMgacaGG0bWaaeWaaeaacaWH4bWaa0 baaSqaaiaadMgacaaMi8UaaiilaiaaykW7caqGVbGaaeyDaiaabsha aeaacaWGubaaaOGabCOSdyaajaaacaGLOaGaayzkaaGaaiOlaaaa@4E67@

Comme il repose sur des dénombrements d’échantillon sans pondération, le modèle (3.1) implique que l’échantillonnage dans les domaines n’est pas informatif compte tenu des valeurs des variables de contexte et des effets aléatoires de domaine. C’est pourquoi le prédicteur (3.3) ne tient pas compte du plan d’échantillonnage complexe. Si le plan de sondage est informatif et qu’on dispose de chiffres pondérés d’enquête, deux grands problèmes se posent. D’abord, les chiffres en question ne seront pas nécessairement les entiers 0 , 1 , , n i ; MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGimaiaacY cacaaMe8UaaGymaiaacYcacaaMe8UaeSOjGSKaaiilaiaaysW7caWG UbWaaSbaaSqaaiaadMgaaeqaaOGaai4oaaaa@421B@ ils prendront plutôt une valeur dans un ensemble fini de nombres en espacement inégal (non nécessairement des entiers) déterminés par la pondération d’enquête des cas de l’échantillon dans le domaine i . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAaiaac6 caaaa@3797@ Ensuite, la variance d’échantillonnage estimée des chiffres pondérés d’enquête y s i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEamaaBa aaleaacaWGZbGaamyAaaqabaaaaa@3907@ qu’implique la distribution binomiale, v ( y s i w ) n i p i w ( 1 p i w ) , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODamaabm aabaGaamyEamaaBaaaleaacaWGZbGaamyAaiaadEhaaeqaaaGccaGL OaGaayzkaaGaeyisISRaamOBamaaBaaaleaacaWGPbaabeaakiaadc hadaWgaaWcbaGaamyAaiaadEhaaeqaaOWaaeWaaeaacaaIXaGaeyOe I0IaamiCamaaBaaaleaacaWGPbGaam4DaaqabaaakiaawIcacaGLPa aacaGGSaaaaa@4A64@ sera erronée. Korn et Graubard (1998) proposent plutôt de modéliser l’estimation pondérée de probabilité d’enquête pour un domaine comme une proportion binomiale avec une « taille d’échantillon efficace » faisant correspondre la variance binomiale résultante à la variance d’échantillonnage réelle de l’estimation directe pondérée d’enquête du domaine. Le recours à une « taille d’échantillon efficace » a été examiné par divers auteurs, dont Mercer et coll. (2014) et Liu et coll. (2014), comme moyen d’intégration de la pondération d’enquête. Mercer et coll. (2014) font observer que le traitement en pseudovraisemblance et le traitement par taille d’échantillon efficace mènent à des estimations identiques des proportions de petits domaines. Si on utilise la taille d’échantillon efficace au lieu de la taille réelle, on se trouve à introduire la pondération d’enquête d’un échantillonnage complexe. Qui plus est, une estimation peut être d’une plus grande précision dans un échantillon complexe que dans un échantillon aléatoire simple, puisqu’on exploite mieux les données de population par un échantillon représentatif issu d’un plan d’échantillonnage approprié. Ici, nous employons l’indice ( e ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaaca WGLbaacaGLOaGaayzkaaaaaa@386A@ pour toutes les quantités à « taille d’échantillon efficace ». Nous abordons les deux problèmes évoqués en définissant une « taille d’échantillon efficace » n i ( e ) , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBamaaBa aaleaacaWGPbWaaeWaaeaacaWGLbaacaGLOaGaayzkaaaabeaakiaa cYcaaaa@3B31@ et un « chiffre d’échantillon efficace » y i s ( e ) , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEamaaBa aaleaacaWGPbGaam4CamaabmaabaGaamyzaaGaayjkaiaawMcaaaqa baGccaGGSaaaaa@3C34@ de sorte que y i s ( e ) = n i ( e ) p i w . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEamaaBa aaleaacaWGPbGaam4CamaabmaabaGaamyzaaGaayjkaiaawMcaaaqa baGccqGH9aqpcaWGUbWaaSbaaSqaaiaadMgadaqadaqaaiaadwgaai aawIcacaGLPaaaaeqaaOGaamiCamaaBaaaleaacaWGPbGaam4Daaqa baGccaGGUaaaaa@44DB@ Cela donne p i w = y i s ( e ) n i ( e ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCamaaBa aaleaacaWGPbGaam4DaaqabaGccqGH9aqpdaWcbaWcbaGaamyEamaa BaaameaacaWGPbGaam4CamaabmaabaGaamyzaaGaayjkaiaawMcaaa qabaaaleaacaWGUbWaaSbaaWqaaiaadMgadaqadaqaaiaadwgaaiaa wIcacaGLPaaaaeqaaaaaaaa@443E@ avec son estimateur correspondant de variance v ( p i w ) . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWG2bWaaeWaaeaapaGaamiCamaaBaaaleaacaWGPbGaam4Daaqa baaak8qacaGLOaGaayzkaaGaaiOlaaaa@3C81@ Nous appliquons ensuite le modèle (3.1) en posant que le chiffre d’échantillon efficace y i s ( e ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEamaaBa aaleaacaWGPbGaam4CamaabmaabaGaamyzaaGaayjkaiaawMcaaaqa baaaaa@3B7A@ dans le petit domaine  i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAaaaa@36E5@ suit une distribution binomiale, c’est-à-dire que y i s ( e ) Bin ( n i ( e ) , π i ) . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEamaaBa aaleaacaWGPbGaam4CamaabmaabaGaamyzaaGaayjkaiaawMcaaaqa baqeeuuDJXwAKbsr4rNCHbacfaGccqWF8iIocaqGcbGaaeyAaiaab6 gadaqadaqaaiaad6gadaWgaaWcbaGaamyAamaabmaabaGaamyzaaGa ayjkaiaawMcaaaqabaGccaGGSaGaaGjbVlabec8aWnaaBaaaleaaca WGPbaabeaaaOGaayjkaiaawMcaaiaac6caaaa@4FC2@ La « taille d’échantillon efficace » n i ( e ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBamaaBa aaleaacaWGPbWaaeWaaeaacaWGLbaacaGLOaGaayzkaaaabeaaaaa@3A77@ est donnée par n i ( e ) = P ^ i ( 1 P ^ i ) v * ( p i w ) , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBamaaBa aaleaacaWGPbWaaeWaaeaacaWGLbaacaGLOaGaayzkaaaabeaakiab g2da9maaleaaleaaceWGqbGbaKaaqaaaaaaaaaWdbmaaBaaameaaca WGPbaabeaalmaabmaabaGaaGymaiabgkHiT8aaceWGqbGbaKaapeWa aSbaaWqaaiaadMgaaeqaaaWccaGLOaGaayzkaaaapaqaaiaadAhada ahaaadbeqaaiaacQcaaaWcdaqadaqaaiaadchadaWgaaadbaGaamyA aiaadEhaaeqaaaWccaGLOaGaayzkaaaaaOGaaiilaaaa@4A74@ P ^ i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmiuayaaja aeaaaaaaaaa8qadaWgaaWcbaGaamyAaaqabaaaaa@3816@ est une prédiction préliminaire par modèle de la proportion de population P i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiuamaaBa aaleaacaWGPbaabeaaaaa@37E6@ dans un modèle linéaire généralisé et où l’estimation de variance v * ( p i w ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODamaaCa aaleqabaGaaiOkaaaakmaabmaabaGaamiCamaaBaaaleaacaWGPbGa am4DaaqabaaakiaawIcacaGLPaaaaaa@3C75@ dépend de P ^ i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmiuayaaja aeaaaaaaaaa8qadaWgaaWcbaGaamyAaaqabaaaaa@3816@ par une fonction de variance généralisée (FVG) (voir Liu et coll. (2014)). Là, y i s ( e ) = 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEamaaBa aaleaacaWGPbGaam4CamaabmaabaGaamyzaaGaayjkaiaawMcaaaqa baGccqGH9aqpcaaIWaaaaa@3D44@ si p i w = 0 , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCamaaBa aaleaacaWGPbGaam4DaaqabaGccqGH9aqpcaaIWaGaaiilaaaa@3B7C@ mais cela ne pose aucun problème, puisque P ^ i > 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmiuayaaja aeaaaaaaaaa8qadaWgaaWcbaGaamyAaaqabaGccqGH+aGpcaaIWaaa aa@39E2@ implique que n i ( e ) > 0. MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBamaaBa aaleaacaWGPbWaaeWaaeaacaWGLbaacaGLOaGaayzkaaaabeaakiab g6da+iaaicdacaGGUaaaaa@3CF5@ À noter que nous employons une fonction de variance généralisée pour produire des estimations de la variance d’échantillonnage même dans le cas des domaines dont le chiffre de dénombrement observé est nul. De la sorte, nous n’excluons aucun domaine de l’ajustement du modèle. Nous obtenons finalement le prédicteur empirique de y i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEamaaBa aaleaacaWGPbaabeaaaaa@380F@ en remplaçant ( n i , y s i ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qadaqadaqaaiaad6gadaWgaaWcbaGaamyAaaqabaGccaGGSaGaaGjb V=aacaWG5bWaaSbaaSqaaiaadohacaWGPbaabeaaaOWdbiaawIcaca GLPaaaaaa@3F2D@ par ( n i ( e ) , y i s ( e ) ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaaca WGUbWaaSbaaSqaaiaadMgadaqadaqaaiaadwgaaiaawIcacaGLPaaa aeqaaOGaaiilaiaaysW7caWG5bWaaSbaaSqaaiaadMgacaWGZbWaae WaaeaacaWGLbaacaGLOaGaayzkaaaabeaaaOGaayjkaiaawMcaaaaa @43D4@ dans (3.2), ce qui garantit que la pondération d’échantillonnage sert à l’estimation sur petits domaines. Plus précisément, le prédicteur plug-in empirique (PE) de y i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEamaaBa aaleaacaWGPbaabeaaaaa@380F@ est alors

y ^ i ( e ) PE = y i s ( e ) + ( N i n i ( e ) ) π ^ i ( e ) PE , ( 3.4 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmyEayaaja Waa0baaSqaaiaadMgadaqadaqaaiaadwgaaiaawIcacaGLPaaaaeaa caqGqbGaaeyraaaakiabg2da9iaadMhadaWgaaWcbaGaamyAaiaado hadaqadaqaaiaadwgaaiaawIcacaGLPaaaaeqaaOGaey4kaSYaaeWa aeaacaWGobWaaSbaaSqaaiaadMgaaeqaaOGaeyOeI0IaamOBamaaBa aaleaacaWGPbWaaeWaaeaacaWGLbaacaGLOaGaayzkaaaabeaaaOGa ayjkaiaawMcaaiqbec8aWzaajaWaa0baaSqaaiaadMgadaqadaqaai aadwgaaiaawIcacaGLPaaaaeaacaqGqbGaaeyraaaakiaacYcacaaM f8UaaGzbVlaaywW7caaMf8UaaGzbVlaacIcacaaIZaGaaiOlaiaais dacaGGPaaaaa@5F9F@

avec π ^ i ( e ) PE = expit ( x i T β ^ ( e ) + z i T u ^ ( e ) ) . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiWdaNbaK aadaqhaaWcbaGaamyAamaabmaabaGaamyzaaGaayjkaiaawMcaaaqa aiaabcfacaqGfbaaaOGaeyypa0JaciyzaiaacIhacaGGWbGaaiyAai aacshadaqadaqaaiaahIhadaqhaaWcbaGaamyAaaqaaiaadsfaaaGc ceWHYoGbaKaadaWgaaWcbaWaaeWaaeaacaWGLbaacaGLOaGaayzkaa aabeaakiabgUcaRiaahQhadaqhaaWcbaGaamyAaaqaaiaadsfaaaGc ceWH1bGbaKaadaWgaaWcbaWaaeWaaeaacaWGLbaacaGLOaGaayzkaa aabeaaaOGaayjkaiaawMcaaiaac6caaaa@5388@ L’estimation de la proportion dans le domaine i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAaaaa@36E5@ est P ^ i ( e ) PE = N i 1 y ^ i ( e ) PE . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmiuayaaja Waa0baaSqaaiaadMgadaqadaqaaiaadwgaaiaawIcacaGLPaaaaeaa caqGqbGaaeyraaaakiabg2da9iaad6eadaqhaaWcbaGaamyAaaqaai abgkHiTiaaigdaaaGcceWG5bGbaKaadaqhaaWcbaGaamyAamaabmaa baGaamyzaaGaayjkaiaawMcaaaqaaiaabcfacaqGfbaaaOGaaiOlaa aa@47A8@  Dans ce cas, il s’agit respectivement avec β ^ ( e ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabCOSdyaaja WaaSbaaSqaamaabmaabaGaamyzaaGaayjkaiaawMcaaaqabaaaaa@39E4@ et u ^ i ( e ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmyDayaaja WaaSbaaSqaaiaadMgadaqadaqaaiaadwgaaiaawIcacaGLPaaaaeqa aaaa@3A8E@ des estimations du paramètre des effets fixes et du prédicteur du paramètre des effets aléatoires dans le modèle (3.2) compte tenu d’une « taille d’échantillon efficace » n i ( e ) , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBamaaBa aaleaacaWGPbWaaeWaaeaacaWGLbaacaGLOaGaayzkaaaabeaakiaa cYcaaaa@3B31@ et d’un « chiffre d’échantillon efficace » y i s ( e ) . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEamaaBa aaleaacaWGPbGaam4CamaabmaabaGaamyzaaGaayjkaiaawMcaaaqa baGccaGGUaaaaa@3C36@ De même, l’estimateur synthétique de y i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEamaaBa aaleaacaWGPbaabeaaaaa@380F@ est y ^ i ( e ) SYN = N i π ^ i ( e ) SYN , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmyEayaaja Waa0baaSqaaiaadMgadaqadaqaaiaadwgaaiaawIcacaGLPaaaaeaa caqGtbGaaeywaiaab6eaaaGccqGH9aqpcaWGobWaaSbaaSqaaiaadM gaaeqaaOGafqiWdaNbaKaadaqhaaWcbaGaamyAamaabmaabaGaamyz aaGaayjkaiaawMcaaaqaaiaabofacaqGzbGaaeOtaaaakiaacYcaaa a@48B5@ avec π ^ i ( e ) SYN = expit ( x i , out T β ^ ( e ) ) . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiWdaNbaK aadaqhaaWcbaGaamyAamaabmaabaGaamyzaaGaayjkaiaawMcaaaqa aiaabofacaqGzbGaaeOtaaaakiabg2da9iGacwgacaGG4bGaaiiCai aacMgacaGG0bWaaeWaaeaacaWH4bWaa0baaSqaaiaadMgacaaMi8Ua aiilaiaaykW7caqGVbGaaeyDaiaabshaaeaacaWGubaaaOGabCOSdy aajaWaaSbaaSqaamaabmaabaGaamyzaaGaayjkaiaawMcaaaqabaaa kiaawIcacaGLPaaacaGGUaaaaa@5383@

3.2  Calcul de la taille d’échantillon efficace

En nous appuyant sur les idées exposées dans Liu et coll. (2014) et Franco et Bell (2013), nous décrivons une procédure permettant de calculer par district les valeurs de taille d’échantillon efficace et le chiffre d’échantillon efficace correspondant pour le dénombrement des ménages pauvres. Nous obtenons d’abord une prédiction approximative par modèle de P i , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiuamaaBa aaleaacaWGPbaabeaakiaacYcaaaa@38A0@ disons P ^ i , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmiuayaaja aeaaaaaaaaa8qadaWgaaWcbaGaamyAaaqabaGccaGGSaaaaa@38D0@ à partir d’un modèle linéaire logistique ajusté à des estimations (pondérées) directes par district p i w MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCamaaBa aaleaacaWGPbGaam4Daaqabaaaaa@3902@ avec un ensemble de variables auxiliaires au niveau des districts. Nous ajustons ce modèle à l’aide de la fonction g l m MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4zaiaadY gacaWGTbaaaa@38C6@ dans le langage R en spécifiant la famille comme « binomiale » et avec les tailles d’échantillon par district comme pondération. Par définition, les estimations par modèle P ^ i = exp ( x i T λ ^ ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmiuayaaja aeaaaaaaaaa8qadaWgaaWcbaGaamyAaaqabaGccqGH9aqppaGaciyz aiaacIhacaGGWbWaaeWaaeaacaWH4bWaa0baaSqaaiaadMgaaeaaca WGubaaaOGabC4UdyaajaaacaGLOaGaayzkaaaaaa@41EF@ se situeront dans l’intervalle (0, 1). Ici, x i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCiEamaaBa aaleaacaWGPbaabeaaaaa@3812@ désigne le vecteur de covariables dans la fonction g l m MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4zaiaadY gacaWGTbaaaa@38C6@ comme dans le modèle (3.1). Nous nous reportons ensuite aux estimations directes de variance s i 2 = v ( p i w ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4CamaaDa aaleaacaWGPbaabaGaaGOmaaaakiabg2da9iaadAhadaqadaqaaiaa dchadaWgaaWcbaGaamyAaiaadEhaaeqaaaGccaGLOaGaayzkaaaaaa@3F6F@ de l’enquête du NSSO comme variables dépendantes dans un modèle FVG. Au moyen des données des districts sans dénombrement nul des ménages pauvres, nous établissons une estimation lissée de la variance d’échantillonnage par le modèle

E ( s i 2 ) = FVG i = α 0 [ P i ( 1 P i ) ] α 1 ( R i ) α 2 . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyramaabm aabaGaam4CamaaDaaaleaacaWGPbaabaGaaGOmaaaaaOGaayjkaiaa wMcaaiabg2da9iaabAeacaqGwbGaae4ramaaBaaaleaacaWGPbaabe aakiabg2da9iabeg7aHnaaBaaaleaacaaIWaaabeaakmaadmaabaGa amiuaabaaaaaaaaapeWaaSbaaSqaaiaadMgaaeqaaOWaaeWaaeaapa GaaGymaiabgkHiTiaadcfadaWgaaWcbaGaamyAaaqabaaak8qacaGL OaGaayzkaaaapaGaay5waiaaw2faamaaCaaaleqabaGaeqySdeMaaG ymaaaakmaabmaabaGaamOuamaaBaaaleaacaWGPbaabeaaaOGaayjk aiaawMcaamaaCaaaleqabaGaeqySdeMaaGOmaaaakiaac6caaaa@5609@

Dans ce cas, R i = Σ j = 1 n i w i j 2 { ( Σ j = 1 n i w i j ) 2 } 1 , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOuamaaBa aaleaacaWGPbaabeaakiabg2da9maavadabeWcbaGaamOAaiabg2da 9iaaigdaaeaacaWGUbWaaSbaaWqaaiaadMgaaeqaaaqdbaGaeu4Odm faaOGaam4DamaaDaaaleaacaWGPbGaamOAaaqaaiaaikdaaaGcdaGa daqaamaabmaabaWaaubmaeqaleaacaWGQbGaeyypa0JaaGymaaqaai aad6gadaWgaaadbaGaamyAaaqabaaaneaacqqHJoWuaaGccaWG3bWa aSbaaSqaaiaadMgacaWGQbaabeaaaOGaayjkaiaawMcaamaaCaaale qabaGaaGOmaaaaaOGaay5Eaiaaw2haamaaCaaaleqabaGaeyOeI0Ia aGymaaaakiaacYcaaaa@5471@ w i j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4DamaaBa aaleaacaWGPbGaamOAaaqabaaaaa@38FC@ est le poids du ménage j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOAaaaa@36E6@ dans le district i . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAaiaac6 caaaa@3797@ Si on prend les logarithmes des deux côtés, on obtient

log FVG i = α 0 * + α 1 log [ P ^ i ( 1 P ^ i ) ] + α 2 log ( R i ) . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciiBaiaac+ gacaGGNbGaaeOraiaabAfacaqGhbWaaSbaaSqaaiaadMgaaeqaaOGa eyypa0JaeqySde2aa0baaSqaaiaaicdaaeaacaGGQaaaaOGaey4kaS IaeqySde2aaSbaaSqaaiaaigdaaeqaaOGaciiBaiaac+gacaGGNbWa amWaaeaaceWGqbGbaKaaqaaaaaaaaaWdbmaaBaaaleaacaWGPbaabe aakmaabmaabaWdaiaaigdacqGHsislceWGqbGbaKaapeWaaSbaaSqa aiaadMgaaeqaaaGccaGLOaGaayzkaaaapaGaay5waiaaw2faaiabgU caRiabeg7aHnaaBaaaleaacaaIYaaabeaakiGacYgacaGGVbGaai4z amaabmaabaGaamOuamaaBaaaleaacaWGPbaabeaaaOGaayjkaiaawM caaiaac6caaaa@5AD8@

Ce modèle peut être ajusté avec la fonction l m MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiBaiaad2 gaaaa@37DA@ dans R pour dégager les estimations α ^ 0 * , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqySdeMbaK aadaqhaaWcbaGaaGimaaqaaiaacQcaaaGccaGGSaaaaa@39F5@ α ^ 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqySdeMbaK aadaWgaaWcbaGaaGymaaqabaaaaa@388D@ et α ^ 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqySdeMbaK aadaWgaaWcbaGaaGOmaaqabaaaaa@388E@ des coefficients de régression. Nous calculons ensuite les estimations FVG lissées de la variance d’échantillonnage comme s ^ i 2 = v * ( p i w ) = exp ( α ^ 0 * ) × [ P ^ i ( 1 P ^ i ) ] α ^ 1 × ( R i ) α ^ 2 . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabm4Cayaaja Waa0baaSqaaiaadMgaaeaacaaIYaaaaOGaeyypa0JaamODamaaCaaa leqabaGaaiOkaaaakmaabmaabaGaamiCamaaBaaaleaacaWGPbGaam 4DaaqabaaakiaawIcacaGLPaaaqaaaaaaaaaWdbiabg2da98aaciGG LbGaaiiEaiaacchadaqadaqaaiqbeg7aHzaajaWaa0baaSqaaiaaic daaeaacaGGQaaaaaGccaGLOaGaayzkaaGaey41aq7aamWaaeaaceWG qbGbaKaapeWaaSbaaSqaaiaadMgaaeqaaOWaaeWaaeaapaGaaGymai abgkHiTiqadcfagaqca8qadaWgaaWcbaGaamyAaaqabaaakiaawIca caGLPaaaa8aacaGLBbGaayzxaaWaaWbaaSqabeaacuaHXoqygaqcam aaBaaameaacaaIXaaabeaaaaGccqGHxdaTdaqadaqaaiaadkfadaWg aaWcbaGaamyAaaqabaaakiaawIcacaGLPaaadaahaaWcbeqaaiqbeg 7aHzaajaWaaSbaaWqaaiaaikdaaeqaaaaakiaaygW7caGGUaaaaa@6249@ Enfin, nous calculons pour chaque district la taille d’échantillon efficace n i ( e ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBamaaBa aaleaacaWGPbWaaeWaaeaacaWGLbaacaGLOaGaayzkaaaabeaaaaa@3A77@ et le chiffre d’échantillon efficace de dénombrement des ménages pauvres y i s ( e ) , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEamaaBa aaleaacaWGPbGaam4CamaabmaabaGaamyzaaGaayjkaiaawMcaaaqa baGccaGGSaaaaa@3C34@ comme n i ( e ) = P ^ i ( 1 P ^ i ) v * ( p i w ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBamaaBa aaleaacaWGPbWaaeWaaeaacaWGLbaacaGLOaGaayzkaaaabeaakiab g2da9maaleaaleaaceWGqbGbaKaaqaaaaaaaaaWdbmaaBaaameaaca WGPbaabeaalmaabmaabaGaaGymaiabgkHiT8aaceWGqbGbaKaapeWa aSbaaWqaaiaadMgaaeqaaaWccaGLOaGaayzkaaaapaqaaiaadAhada ahaaadbeqaaiaacQcaaaWcdaqadaqaaiaadchadaWgaaadbaGaamyA aiaadEhaaeqaaaWccaGLOaGaayzkaaaaaaaa@49BA@ et y i s ( e ) = n i ( e ) p i w . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEamaaBa aaleaacaWGPbGaam4CamaabmaabaGaamyzaaGaayjkaiaawMcaaaqa baGccqGH9aqpcaWGUbWaaSbaaSqaaiaadMgadaqadaqaaiaadwgaai aawIcacaGLPaaaaeqaaOGaamiCamaaBaaaleaacaWGPbGaam4Daaqa baGccaGGUaaaaa@44DB@ Nous arrondissons les valeurs à l’entier le plus proche. La figure 3.1 présente les tailles d’échantillon efficaces par rapport aux tailles d’échantillon observées. Les chiffres d’échantillon efficaces et observés se trouvent à la figure 3.2. Dans la majorité des cas, la taille efficace est inférieure à la taille observée. De même, le chiffre d’échantillon efficace est le plus souvent inférieur au chiffre observé, indice que le plan de sondage fait perdre de l’information si on le compare pour les mêmes districts à un échantillonnage aléatoire simple. La figure 3.3 livre par district les estimations directes d’enquête, pondérées et non pondérées, de la proportion de ménages pauvres. Il ressort de cette figure que les estimations directes non pondérées sous-évaluent la proportion de ménages pauvres.

Figure 3.1 Taille d’échantillon efficace et taille d’échantillon observée pour les données du NSSO

Description de la figure 3.1 

Graphique en nuage de points présentant la taille d’échantillon efficace (axe des y) en fonction de la taille d’échantillon observée (axe des x) pour les données du NSSO. Les données sont dans le tableau suivant : 

Tableau de données de la figure 3.1
Sommaire du tableau
Le tableau montre les résultats de Tableau de données de la figure 3.1. Les données sont présentées selon Taille d'échantillon observée (titres de rangée) et Taille d'échantillon efficace(figurant comme en-tête de colonne).
Taille d'échantillon observée Taille d'échantillon efficace
32 25
32 24
32 21
32 15
32 13
32 16
32 38
63 34
63 30
64 38
64 34
64 43
64 33
64 45
64 42
64 35
64 36
64 26
64 35
64 41
64 35
64 31
64 28
64 35
64 30
64 33
64 35
64 38
64 27
64 30
64 27
64 45
64 27
64 40
64 39
64 36
95 31
95 47
96 76
96 53
96 89
96 45
96 42
96 41
96 50
96 47
96 39
96 48
96 36
96 48
96 47
96 41
96 44
96 36
96 60
96 51
128 67
128 60
128 46
128 61
128 59
128 68
128 52
128 46
128 70
128 52
128 66
128 53
128 59
128 51
128 61

Figure 3.2 Chiffre d’échantillon efficace et chiffre d’échantillon observé pour les données du NSSO

Description de la figure 3.2 

Graphique en nuage de points présentant le chiffre d’échantillon efficace (axe des y) en fonction du chiffre d’échantillon observé (axe des x) pour les données du NSSO. Les données sont dans le tableau suivant : 

Tableau de données de la figure 3.2
Sommaire du tableau
Le tableau montre les résultats de Tableau de données de la figure 3.2. Les données sont présentées selon Chiffre d'échantillon observé (titres de rangée) et Chiffre d'échantillon efficace(figurant comme en-tête de colonne).
Chiffre d'échantillon observé Chiffre d'échantillon efficace
3 4
1 0
4 3
7 3
4 5
9 3
5 6
29 15
12 6
15 9
12 9
1 0
3 2
2 0
8 8
14 9
6 5
19 12
8 6
12 14
11 6
12 10
6 3
10 5
7 4
12 4
7 7
6 5
26 13
17 9
20 10
19 15
12 4
12 8
26 20
18 14
13 6
5 2
5 5
18 9
8 9
12 9
16 10
18 11
38 28
38 24
29 20
30 15
29 18
30 12
46 27
36 14
26 15
26 10
18 11
25 12
8 3
24 8
36 14
40 20
38 15
42 25
48 23
34 11
27 15
37 14
34 18
29 11
39 19
24 9
29 15

Figure 3.3 Estimations directes d’enquête pondérées et non pondérées par district de la proportion de ménages pauvres

Description de la figure 3.3 

Graphique en nuage de points présentant les estimations directes d’enquête pondérées (axe des y) et non pondérées (axe des x) par district de la proportion de ménages pauvres. Les données sont dans le tableau suivant : 

Tableau de données de la figure 3.3
Sommaire du tableau
Le tableau montre les résultats de Tableau de données de la figure 3.3. Les données sont présentées selon Estimations directes non pondérées (titres de rangée) et Estimations directes d'enquête pondérées(figurant comme en-tête de colonne).
Estimations directes non pondérées Estimations directes d'enquête pondérées
0,09 0,14
0,03 0,02
0,13 0,14
0,22 0,17
0,13 0,35
0,28 0,20
0,16 0,16
0,46 0,45
0,19 0,20
0,23 0,24
0,19 0,27
0,02 0,00
0,05 0,05
0,03 0,01
0,13 0,18
0,22 0,25
0,09 0,13
0,30 0,45
0,13 0,18
0,19 0,35
0,17 0,18
0,19 0,31
0,09 0,09
0,16 0,15
0,11 0,15
0,19 0,12
0,11 0,21
0,09 0,12
0,41 0,49
0,27 0,29
0,31 0,36
0,30 0,33
0,19 0,15
0,19 0,19
0,41 0,51
0,28 0,38
0,14 0,18
0,05 0,05
0,05 0,07
0,19 0,17
0,08 0,10
0,13 0,19
0,17 0,23
0,19 0,27
0,40 0,57
0,40 0,52
0,30 0,50
0,31 0,31
0,30 0,49
0,31 0,26
0,48 0,58
0,38 0,35
0,27 0,35
0,27 0,27
0,19 0,19
0,26 0,24
0,06 0,05
0,19 0,13
0,28 0,30
0,31 0,32
0,30 0,26
0,33 0,37
0,38 0,45
0,27 0,24
0,21 0,21
0,29 0,27
0,27 0,28
0,23 0,21
0,31 0,32
0,19 0,18
0,23 0,24

3.3  Effets aléatoires de domaine en corrélation spatiale

Dans le modèle (3.2), on suppose que les effets aléatoires de domaine ne sont pas en corrélation. Dans de nombreuses applications, la situation des domaines dans l’espace est le reflet d’une information de contexte manquante et l’hypothèse d’absence de corrélation est à mettre en doute dans le cas des effets aléatoires de domaine. Il est souvent raisonnable de supposer que les effets de domaines voisins (définis, par exemple, par un critère de contiguïté) sont en corrélation et que la corrélation décroît jusqu’à zéro à mesure qu’augmente la distance entre les domaines. Pour tenir compte de la dépendance entre domaines voisins, on emploie souvent dans un traitement d’EPD des modèles spatiaux pour les effets aléatoires de domaine (Pratesi et Salvati, 2008; Chandra et Salvati, 2018). Nous le faisons en introduisant une dépendance spatiale dans la structure des erreurs du modèle (3.2). Plus précisément, nous posons un processus d’autorégression simultanée (ARS) des erreurs (Pratesi et Salvati, 2008) où le vecteur des effets aléatoires de domaine v = ( v i ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbeGaa8NDai abg2da9maabmaabaGaamODamaaBaaaleaacaWGPbaabeaaaOGaayjk aiaawMcaaaaa@3BA6@ peut s’exprimer sous la forme

v = ρ L v + u , ( 3.5 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCODaiabg2 da9iabeg8aYjaahYeacaWH2bGaey4kaSIaaCyDaiaacYcacaaMf8Ua aGzbVlaaywW7caaMf8UaaGzbVlaacIcacaaIZaGaaiOlaiaaiwdaca GGPaaaaa@496C@

v = ( I D ρ L ) 1 u , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCODaiabg2 da9maabmaabaGaaCysamaaBaaaleaaieGacaWFebaabeaakiabgkHi Tiabeg8aYjaahYeaaiaawIcacaGLPaaadaahaaWcbeqaaiabgkHiTi aaigdaaaGccaWH1bGaaiilaaaa@426C@ avec E ( v ) = 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyramaabm aabaGaaCODaaGaayjkaiaawMcaaiabg2da9iaaicdaaaa@3B09@ et Var ( v ) = σ u 2 [ ( I D ρ L ) ( I D ρ L T ) ] 1 = Ω s p ( δ ) . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeOvaiaabg gacaqGYbWaaeWaaeaacaWH2baacaGLOaGaayzkaaGaeyypa0Jaeq4W dm3aa0baaSqaaiaadwhaaeaacaaIYaaaaOWaamWaaeaadaqadaqaai aahMeadaWgaaWcbaGaamiraaqabaGccqGHsislcqaHbpGCcaWHmbaa caGLOaGaayzkaaWaaeWaaeaacaWHjbWaaSbaaSqaaiaadseaaeqaaO GaeyOeI0IaeqyWdiNaaCitamaaCaaaleqabaGaamivaaaaaOGaayjk aiaawMcaaaGaay5waiaaw2faamaaCaaaleqabaGaeyOeI0IaaGymaa aakiabg2da9iaahM6adaWgaaWcbaGaam4CaiaadchaaeqaaOWaaeWa aeaacaWH0oaacaGLOaGaayzkaaGaaiOlaaaa@5A58@ Ici, L MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCitaaaa@36CC@ est une matrice de proximité d’ordre D , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiraiaacY caaaa@3770@ u N ( 0 , σ u 2 I D ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCyDaebbfv 3ySLgzGueE0jxyaGqbaiab=XJi6iaad6eadaqadaqaaiaaicdacaGG SaGaaGjbVlabeo8aZnaaDaaaleaacaWG1baabaGaaGOmaaaakiaahM eadaWgaaWcbaGaamiraaqabaaakiaawIcacaGLPaaaaaa@4782@ et ρ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyWdihaaa@37B7@ est un coefficient d’autorégression spatiale. À partir de (3.2) et (3.5), le MLGM en dépendance spatiale (MLGMS) s’obtient par la forme

η = X β + Z ( I D ρ L ) 1 u = X β + Z v . ( 3.6 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaC4Tdiabg2 da9iaahIfacaWHYoGaey4kaSIaaCOwamaabmaabaGaaCysamaaBaaa leaaieGacaWFebaabeaakiabgkHiTiabeg8aYjaahYeaaiaawIcaca GLPaaadaahaaWcbeqaaiabgkHiTiaaigdaaaGccaWH1bGaeyypa0Ja aCiwaiaahk7acqGHRaWkcaWHAbGaaCODaiaac6cacaaMf8UaaGzbVl aaywW7caaMf8UaaGzbVlaacIcacaaIZaGaaiOlaiaaiAdacaGGPaaa aa@57CC@

La matrice de proximité L MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCitaaaa@36CC@ décrit comment les effets aléatoires des domaines voisins sont liés, alors que ρ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyWdihaaa@37B7@ définit l’étroitesse de ce rapport spatial. En d’autres termes, c’est l’indicateur du degré de similitude entre un objet et les objets voisins. Dans ce qui suit, nous employons l’indice « sp » pour désigner les quantités liées dans le modèle (3.5). La forme la plus simple pour définir L MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCitaaaa@36CC@ est une matrice de contiguïté, c’est-à-dire une matrice carrée binaire L MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCitaaaa@36CC@ d’ordre D MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiraaaa@36C0@ avec des valeurs non nulles seulement pour les paires de domaines adjacents. Pour faciliter l’interprétation, nous prêtons généralement à cette matrice une forme réduite et centrée de ligne, auquel cas ρ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyWdihaaa@37B7@ est appelé paramètre d’autocorrélation spatiale (Banerjee, Carlin et Gelfand, 2004). L’élément l j k ( j , k = 1 , , D ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiBamaaBa aaleaacaWGQbGaam4AaaqabaGcdaqadaqaaiaadQgacaGGSaGaaGjb VlaadUgacqGH9aqpcaaIXaGaaiilaiaaysW7cqWIMaYscaGGSaGaaG jbVlaadseaaiaawIcacaGLPaaaaaa@46C8@ d’une matrice de contiguïté prend formellement la valeur 1 si le domaine j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOAaaaa@36E6@ a une arête commune avec le domaine k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4Aaaaa@36E7@ et 0 dans les autres cas. Sous une forme réduite et centrée de ligne, cela devient

l j k = { t j 1 si j  et  k  sont contigus 0 dans les autres cas, ( 3.7 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiBamaaBa aaleaacaWGQbGaam4AaaqabaGccqGH9aqpdaGabaqaauaabaqaciaa aeaacaWG0bWaa0baaSqaaiaadQgaaeaacqGHsislcaaIXaaaaaGcba Gaae4CaiaabMgacaaMe8UaamOAaiaabccacaqGLbGaaeiDaiaabcca caWGRbGaaeiiaiaabohacaqGVbGaaeOBaiaabshacaaMe8Uaae4yai aab+gacaqGUbGaaeiDaiaabMgacaqGNbGaaeyDaiaabohaaeaacaaI WaaabaGaaeizaiaabggacaqGUbGaae4CaiaabccacaqGSbGaaeyzai aabohacaqGGaGaaeyyaiaabwhacaqG0bGaaeOCaiaabwgacaqGZbGa aeiiaiaabogacaqGHbGaae4CaiaabYcaaaGaaGzbVlaaywW7caaMf8 UaaGzbVlaaywW7caGGOaGaaG4maiaac6cacaaI3aGaaiykaaGaay5E aaaaaa@726F@

t j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiDamaaBa aaleaacaWGQbaabeaaaaa@380B@ est le nombre total de domaines partageant une arête avec le domaine j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOAaaaa@36E6@ (ce qui comprend le domaine j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOAaaaa@36E6@ lui-même). La contiguïté représente sans doute la spécification la plus simple, mais non nécessairement la meilleure, d’une matrice d’interaction spatiale (voir Chandra (2013), par exemple). Il est peut-être plus informatif que cette interaction soit précisée, par exemple, comme une certaine fonction de la longueur d’une limite commune entre domaines voisins ou de la distance entre domaines. Pour notre propos, nous examinerons donc une définition fondée sur la distance pour la matrice de proximité L MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCitaaaa@36CC@ (matrice dont les entrées sont fonction de la distance entre les petits domaines ou districts). Nous nous attachons aux différentes façons de définir la proximité (ou la matrice de pondération spatiale), le but étant de reconnaître le meilleur moyen d’exploiter l’information spatiale pour produire des estimations fiables de petits domaines. Disons que la situation dans l’espace du domaine ou du district i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAaaaa@36E5@ correspond aux coordonnées d’un point spatial arbitrairement défini (coordonnées x y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabgk HiTiaadMhaaaa@38DE@ ou longitude et latitude dans une représentation bidimensionnelle) dans un domaine (centroïde, par exemple), ce que nous désignerons par c i = ( c i x , c i y ) . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4yamaaBa aaleaacaWGPbaabeaakiabg2da9maabmaabaGaam4yamaaBaaaleaa caWGPbGaamiEaaqabaGccaGGSaGaaGjbVlaadogadaWgaaWcbaGaam yAaiaadMhaaeqaaaGccaGLOaGaayzkaaGaaiOlaaaa@4394@ Soit d j k = c j c k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamizamaaBa aaleaacaWGQbGaam4AaaqabaGccqGH9aqpdaqbdaqaaiaaykW7caWG JbWaaSbaaSqaaiaadQgaaeqaaOGaeyOeI0Iaam4yamaaBaaaleaaca WGRbaabeaakiaaykW7aiaawMa7caGLkWoaaaa@4540@ une mesure appropriée de la distance entre les points spatiaux des domaines j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOAaaaa@36E6@ et i . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAaiaac6 caaaa@3797@ Nous considérons alors les spécifications suivantes de la matrice de proximité L MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCitaaaa@36CC@ en fonction de la distance :

(i)
Proximité définie comme l’inverse de la distance entre les domaines :

L = { l j k } = { d j k 1 j k ; 0 j = k . ( 3.8 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCitaiabg2 da9maacmaabaGaamiBamaaBaaaleaacaWGQbGaam4Aaaqabaaakiaa wUhacaGL9baacqGH9aqpdaGabaqaauaabaqaciaaaeaacaWGKbWaa0 baaSqaaiaadQgacaWGRbaabaGaeyOeI0IaaGymaaaaaOqaaiaadQga cqGHGjsUcaWGRbGaai4oaaqaaiaaicdaaeaacaWGQbGaeyypa0Jaam 4Aaiaac6caaaaacaGL7baacaaMf8UaaGzbVlaaywW7caaMf8UaaGzb VlaacIcacaaIZaGaaiOlaiaaiIdacaGGPaaaaa@57E4@

(ii)
Spécification exponentielle de la fonction de proximité ainsi définie :

L = { l j k } = exp { 0,5 d j k 2 } ( 3.9 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCitaiabg2 da9maacmaabaGaamiBamaaBaaaleaacaWGQbGaam4Aaaqabaaakiaa wUhacaGL9baacqGH9aqpciGGLbGaaiiEaiaacchadaGadaqaaiabgk HiTiaabcdacaqGSaGaaeynaiaadsgadaqhaaWcbaGaamOAaiaadUga aeaacaaIYaaaaaGccaGL7bGaayzFaaGaaGzbVlaaywW7caaMf8UaaG zbVlaaywW7caGGOaGaaG4maiaac6cacaaI5aGaaiykaaaa@552D@

(iii)
Spécification gaussienne de la fonction de proximité ainsi définie :

L = { l j k } = exp { 0,5 ( d j k / b ) 2 } , ( 3.10 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCitaiabg2 da9maacmaabaGaamiBamaaBaaaleaacaWGQbGaam4Aaaqabaaakiaa wUhacaGL9baacqGH9aqpciGGLbGaaiiEaiaacchadaGadaqaaiabgk HiTiaabcdacaqGSaGaaeynamaabmaabaWaaSGbaeaacaWGKbWaaSba aSqaaiaadQgacaWGRbaabeaaaOqaaiaadkgaaaaacaGLOaGaayzkaa WaaWbaaSqabeaacaaIYaaaaaGccaGL7bGaayzFaaGaaiilaiaaywW7 caaMf8UaaGzbVlaaywW7caaMf8UaaiikaiaaiodacaGGUaGaaGymai aaicdacaGGPaaaaa@594B@

où le paramètre b MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOyaaaa@36DE@ est la largeur de bande, laquelle peut se définir au mieux à l’aide du critère des moindres carrés (Fotheringham, Brunsdon et Charlton, 2002). La largeur de bande est une mesure de la vitesse de décroissance de la proximité à mesure que croît la distance. Nous employons une procédure d’intervalidation pour estimer cette largeur. Dans ce cas, la proximité en (3.10) diminue exponentiellement à mesure qu’augmente la distance entre les domaines j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOAaaaa@36E6@ et k . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4Aaiaac6 caaaa@3799@ Plus particulièrement, nous employons la fonction g w r . s e l MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4zaiaadE hacaWGYbGaaiOlaiaadohacaWGLbGaamiBaaaa@3C5B@ dans le paquet s p g w r MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4Caiaadc hacaWGNbGaam4Daiaadkhaaaa@3AC3@ de R pour calculer la valeur de largeur de bande. Nous utilisons également la fonction M o r a n . I MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamytaiaad+ gacaWGYbGaamyyaiaad6gacaGGUaGaamysaaaa@3C0D@ du paquet a p e MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaiaadc hacaWGLbaaaa@38BC@ de R pour vérifier la présence d’une corrélation spatiale dans les données. Selon les résultats de cette analyse, les données du NSSO présentent une certaine autocorrélation spatiale. Nous rejetons en particulier l’hypothèse nulle d’absence de corrélation spatiale au niveau de signification de 1 %.

Dans le modèle (3.5), il y a deux paramètres ( σ u 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiabeo 8aZnaaDaaaleaacaWG1baabaGaaGOmaaaaaaa@3A49@ et ρ ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyWdiNaai ykaaaa@3864@ à estimer. Posons que δ = ( σ u 2 , ρ ) T . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCiTdiabg2 da9maabmaabaGaeq4Wdm3aa0baaSqaaiaadwhaaeaacaaIYaaaaOGa aiilaiaaysW7cqaHbpGCaiaawIcacaGLPaaadaahaaWcbeqaaiaads faaaGccaGGUaaaaa@4335@ Si nous remplaçons ces paramètres inconnus par leurs valeurs estimées δ ^ = ( σ ^ u 2 , ρ ^ ) T , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabCiTdyaaja Gaeyypa0ZaaeWaaeaacuaHdpWCgaqcamaaDaaaleaacaWG1baabaGa aGOmaaaakiaacYcacaaMe8UafqyWdiNbaKaaaiaawIcacaGLPaaada ahaaWcbeqaaiaadsfaaaGccaaMb8Uaaiilaaaa@44ED@ et désignons les estimateurs plug-in qui suivent par un accent circonflexe, nous définissons le prédicteur empirique spatial (PES) du chiffre de population dans le domaine i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAaaaa@36E5@ par

y ^ i ( e ) PES = y i s ( e ) + ( N i n i ( e ) ) π ^ i ( e ) PES , ( 3.11 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmyEayaaja Waa0baaSqaaiaadMgadaqadaqaaiaadwgaaiaawIcacaGLPaaaaeaa caqGqbGaaeyraiaabofaaaGccqGH9aqpcaWG5bWaaSbaaSqaaiaadM gacaWGZbWaaeWaaeaacaWGLbaacaGLOaGaayzkaaaabeaakiabgUca RiaaykW7daqadaqaaiaad6eadaWgaaWcbaGaamyAaaqabaGccaaMc8 UaeyOeI0IaamOBamaaBaaaleaacaWGPbWaaeWaaeaacaWGLbaacaGL OaGaayzkaaaabeaaaOGaayjkaiaawMcaaiqbec8aWzaajaWaa0baaS qaaiaadMgadaqadaqaaiaadwgaaiaawIcacaGLPaaaaeaacaqGqbGa aeyraiaabofaaaGccaGGSaGaaGzbVlaaywW7caaMf8UaaGzbVlaayw W7caGGOaGaaG4maiaac6cacaaIXaGaaGymaiaacMcaaaa@6519@

avec π ^ i ( e ) PES = expit ( x i T β ^ ( e ) s p + z i T v ^ ( e ) ) . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiWdaNbaK aadaqhaaWcbaGaamyAamaabmaabaGaamyzaaGaayjkaiaawMcaaaqa aiaabcfacaqGfbGaae4uaaaakiabg2da9iGacwgacaGG4bGaaiiCai aacMgacaGG0bWaaeWaaeaacaWH4bWaa0baaSqaaiaadMgaaeaacaWG ubaaaOGabCOSdyaajaWaa0baaSqaamaabmaabaGaamyzaaGaayjkai aawMcaaaqaaiaadohacaWGWbaaaOGaey4kaSIaaCOEamaaDaaaleaa caWGPbaabaGaamivaaaakiqahAhagaqcamaaBaaaleaadaqadaqaai aadwgaaiaawIcacaGLPaaaaeqaaaGccaGLOaGaayzkaaGaaiOlaaaa @564D@ Le prédicteur empirique spatial correspondant de la proportion dans le domaine i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAaaaa@36E5@ est P ^ i ( e ) PES = N i 1 y ^ i ( e ) PES . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmiuayaaja Waa0baaSqaaiaadMgadaqadaqaaiaadwgaaiaawIcacaGLPaaaaeaa caqGqbGaaeyraiaabofaaaGccqGH9aqpcaWGobWaa0baaSqaaiaadM gaaeaacqGHsislcaaIXaaaaOGabmyEayaajaWaa0baaSqaaiaadMga daqadaqaaiaadwgaaiaawIcacaGLPaaaaeaacaqGqbGaaeyraiaabo faaaGccaGGUaaaaa@4954@ Dans le cas d’un domaine non échantillonné, le prédicteur empirique spatial synthétique (désigné par SpSyn dans la suite du texte) de P i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiuamaaBa aaleaacaWGPbaabeaaaaa@37E6@ est P ^ i ( e ) SpSyn = expit ( x i , out T β ^ ( e ) s p ) . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmiuayaaja Waa0baaSqaaiaadMgadaqadaqaaiaadwgaaiaawIcacaGLPaaaaeaa caqGtbGaaeiCaiaabofacaqG5bGaaeOBaaaakiabg2da9iGacwgaca GG4bGaaiiCaiaacMgacaGG0bWaaeWaaeaacaWH4bWaa0baaSqaaiaa dMgacaaMb8UaaiilaiaaykW7caqGVbGaaeyDaiaabshaaeaacaWGub aaaOGabCOSdyaajaWaa0baaSqaamaabmaabaGaamyzaaGaayjkaiaa wMcaaaqaaiaadohacaWGWbaaaaGccaGLOaGaayzkaaGaaiOlaaaa@568B@ Là, β ^ ( e ) s p MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabCOSdyaaja Waa0baaSqaamaabmaabaGaamyzaaGaayjkaiaawMcaaaqaaiaadoha caWGWbaaaaaa@3BD2@ est l’estimation des paramètres d’effets fixes β MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCOSdaaa@3735@ et v ^ ( e ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabCODayaaja WaaSbaaSqaamaabmaabaGaamyzaaGaayjkaiaawMcaaaqabaaaaa@39A5@ est la valeur prédite des effets aléatoires en corrélation spatiale v MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCODaaaa@36F6@ dans le modèle ARS avec la « taille d’échantillon efficace » et le « chiffre d’échantillon efficace ». L’estimation des paramètres de modèle inconnus en (3.6) découle de la procédure examinée à la section précédente, mais la composante de variance σ u 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aa0 baaSqaaiaadwhaaeaacaaIYaaaaaaa@399D@ est maintenant δ = ( σ u 2 , ρ ) T MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCiTdiabg2 da9maabmaabaGaeq4Wdm3aa0baaSqaaiaadwhaaeaacaaIYaaaaOGa aiilaiaaysW7cqaHbpGCaiaawIcacaGLPaaadaahaaWcbeqaaiaads faaaaaaa@4279@ et l’effet aléatoire prédit u ^ ( e ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabCyDayaaja WaaSbaaSqaamaabmaabaGaamyzaaGaayjkaiaawMcaaaqabaaaaa@39A4@ est remplacé par v ^ ( e ) . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabCODayaaja WaaSbaaSqaamaabmaabaGaamyzaaGaayjkaiaawMcaaaqabaGccaGG Uaaaaa@3A61@

3.4  Estimation de l’erreur quadratique moyenne (EQM)

L’estimation d’erreur quadratique moyenne du prédicteur empirique de petits domaines (3.4) suit ce que décrivent Saei et Chambers (2003), Manteiga et coll. (2007), Johnson et coll. (2010), Chandra et coll. (2011), Chandra, Salvati et Chambers (2018) et les références présentées. Nous utilisons directement l’estimation EQM du prédicteur empirique (3.4) par l’expression (3.12) en remplaçant respectivement les tailles et les chiffres d’échantillon observés par les tailles et les chiffres efficaces comme pondération d’enquête. Dans le modèle (3.2), une estimation approximative EQM du PE (3.4) par les tailles et les chiffres d’échantillon efficaces est

eqm ( P ^ i ( e ) PE ) m 1 i ( σ ^ u 2 ) + m 2 i ( σ ^ u 2 ) + 2 m 3 i ( σ ^ u 2 ) . ( 3.12 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeyzaiaabg hacaqGTbWaaeWaaeaaceWGqbGbaKaadaqhaaWcbaGaamyAamaabmaa baGaamyzaaGaayjkaiaawMcaaaqaaiaabcfacaqGfbaaaaGccaGLOa GaayzkaaGaeyisISRaamyBamaaBaaaleaacaaIXaGaamyAaaqabaGc daqadaqaaiqbeo8aZzaajaWaa0baaSqaaiaadwhaaeaacaaIYaaaaa GccaGLOaGaayzkaaGaey4kaSIaamyBamaaBaaaleaacaaIYaGaamyA aaqabaGcdaqadaqaaiqbeo8aZzaajaWaa0baaSqaaiaadwhaaeaaca aIYaaaaaGccaGLOaGaayzkaaGaey4kaSIaaGOmaiaad2gadaWgaaWc baGaaG4maiaadMgaaeqaaOWaaeWaaeaacuaHdpWCgaqcamaaDaaale aacaWG1baabaGaaGOmaaaaaOGaayjkaiaawMcaaiaac6cacaaMf8Ua aGzbVlaaywW7caaMf8UaaGzbVlaacIcacaaIZaGaaiOlaiaaigdaca aIYaGaaiykaaaa@699C@

Cette estimation de l’erreur quadratique moyenne est fondée sur une approximation analogue à celle du modèle linéaire mixte (voir Rao et Molina (2015), chapitre 5, pages 100-107), Saei et Chambers (2003) et Manteiga et coll. (2007)). Pour définir trois composantes de cette erreur (3.12), soit Σ ^ = Z T B Z + Ω ^ 1 , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabC4Odyaaja Gaeyypa0JaaCOwamaaCaaaleqabaGaamivaaaakiaahkeacaWHAbGa ey4kaSIabCyQdyaajaWaaWbaaSqabeaacqGHsislcaaIXaaaaOGaai ilaaaa@4093@ X * = { diag ( N i 1 ) } H X MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCiwamaaCa aaleqabaGaaiOkaaaakiabg2da9maacmaabaGaaeizaiaabMgacaqG HbGaae4zamaabmaabaGaamOtamaaDaaaleaacaWGPbaabaGaeyOeI0 IaaGymaaaaaOGaayjkaiaawMcaaaGaay5Eaiaaw2haaiaahIeacaWH ybaaaa@4570@ et Z * = { diag ( N i 1 ) } H Z , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCOwamaaCa aaleqabaGaaiOkaaaakiabg2da9maacmaabaGaaeizaiaabMgacaqG HbGaae4zamaabmaabaGaamOtamaaDaaaleaacaWGPbaabaGaeyOeI0 IaaGymaaaaaOGaayjkaiaawMcaaaGaay5Eaiaaw2haaiaahIeacaWH AbGaaiilaaaa@4624@

H = h ( η ) η | η ^ = η = diag { P ^ i ( e ) PE ( 1 P ^ i ( e ) PE ) ; i = 1 , , D } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCisaiabg2 da9maaeiaabaWaaSaaaeaacqGHciITcaWGObWaaeWaaeaacaWH3oaa caGLOaGaayzkaaaabaGaeyOaIyRaaC4TdaaacaaMc8oacaGLiWoada WgaaWcbaGabC4TdyaajaGaeyypa0JaaC4TdaqabaGccqGH9aqpcaqG KbGaaeyAaiaabggacaqGNbWaaiWaaeaaceWGqbGbaKaadaqhaaWcba GaamyAamaabmaabaGaamyzaaGaayjkaiaawMcaaaqaaiaabcfacaqG fbaaaOWaaeWaaeaacaaIXaGaeyOeI0IabmiuayaajaWaa0baaSqaai aadMgadaqadaqaaiaadwgaaiaawIcacaGLPaaaaeaacaqGqbGaaeyr aaaaaOGaayjkaiaawMcaaiaacUdacaaMe8UaaGPaVlaadMgacqGH9a qpcaaIXaGaaiilaiaaysW7cqWIMaYscaGGSaGaaGjbVlaadseaaiaa wUhacaGL9baaaaa@69BC@

et où

B = 2 l 1 η η T | η ^ = η = diag { n i ( e ) P ^ i ( e ) PE ( 1 P ^ i ( e ) PE ) ; i = 1 , , D } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCOqaiabg2 da9iabgkHiTmaaeiaabaWaaSaaaeaacqaHciITdaahaaWcbeqaaiaa ikdaaaGccaWGSbWaaSbaaSqaaiaaigdaaeqaaaGcbaGaeqOaIyRaaC 4TdiabekGi2kaahE7adaahaaWcbeqaaiaadsfaaaaaaOGaaGPaVdGa ayjcSdWaaSbaaSqaaiqahE7agaqcaiabg2da9iaahE7aaeqaaOGaey ypa0JaaeizaiaabMgacaqGHbGaae4zamaacmaabaGaamOBamaaBaaa leaacaWGPbWaaeWaaeaacaWGLbaacaGLOaGaayzkaaaabeaakiqadc fagaqcamaaDaaaleaacaWGPbWaaeWaaeaacaWGLbaacaGLOaGaayzk aaaabaGaaeiuaiaabweaaaGcdaqadaqaaiaaigdacqGHsislceWGqb GbaKaadaqhaaWcbaGaamyAamaabmaabaGaamyzaaGaayjkaiaawMca aaqaaiaabcfacaqGfbaaaaGccaGLOaGaayzkaaGaai4oaiaaysW7ca aMc8UaamyAaiabg2da9iaaigdacaGGSaGaaGjbVlablAciljaacYca caaMe8UaamiraaGaay5Eaiaaw2haaaaa@71FC@

est la matrice des dérivées secondes de l 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiBamaaBa aaleaacaaIXaaabeaaaaa@37CF@ (fonction de vraisemblance logarithmique l 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiBamaaBa aaleaacaaIXaaabeaaaaa@37CF@ définie par le vecteur y s ( e ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCyEamaaBa aaleaacaWGZbWaaeWaaeaacaWGLbaacaGLOaGaayzkaaaabeaaaaa@3A90@ étant donné u ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCyDaiaacM caaaa@37A2@ pour η MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaC4Tdaaa@373A@ à η = η ^ . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaC4Tdiabg2 da9iqahE7agaqcaiaac6caaaa@3A45@ D’après McGilchrist (1994), nous pouvons former la matrice de variances-covariances comme

V ^ = [ X T Z T ] B [ X Z ] + [ 0 0 0 Ω ^ 1 ] = ( X T B X X T B Z Z T B X Σ ^ ) = ( V ^ 11 V ^ 12 V ^ 21 V ^ 22 ) , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabCOvayaaja Gaeyypa0ZaamWaaeaafaqaaeqabaaabaqbaeaabiqaaaqaaiaahIfa daahaaWcbeqaaiaadsfaaaaakeaacaWHAbWaaWbaaSqabeaacaWGub aaaaaaaaaakiaawUfacaGLDbaacaWHcbWaamWaaeaafaqabeqacaaa baGaaCiwaaqaaiaahQfaaaaacaGLBbGaayzxaaGaey4kaSYaamWaae aafaqaaeGacaaabaGaaCimaaqaaiaahcdaaeaacaWHWaaabaGabCyQ dyaajaWaaWbaaSqabeaacqGHsislcaaIXaaaaaaaaOGaay5waiaaw2 faaiabg2da9maabmaabaqbaeqabiGaaaqaaiaahIfadaahaaWcbeqa aiaadsfaaaGccaWHcbGaaCiwaaqaaiaahIfadaahaaWcbeqaaiaads faaaGccaWHcbGaaCOwaaqaaiaahQfadaahaaWcbeqaaiaadsfaaaGc caWHcbGaaCiwaaqaaiqaho6agaqcaaaaaiaawIcacaGLPaaacqGH9a qpdaqadaqaauaabaqaciaaaeaaceWHwbGbaKaadaWgaaWcbaGaaGym aiaaigdaaeqaaaGcbaGabCOvayaajaWaaSbaaSqaaiaaigdacaaIYa aabeaaaOqaaiqahAfagaqcamaaBaaaleaacaaIYaGaaGymaaqabaaa keaaceWHwbGbaKaadaWgaaWcbaGaaGOmaiaaikdaaeqaaaaaaOGaay jkaiaawMcaaiaacYcaaaa@670E@

de sorte que

V ^ 1 = ( X T B X X T B Z Z T B X Σ ^ ) 1 = ( A ^ 11 A ^ 12 A ^ 21 A ^ 22 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabCOvayaaja WaaWbaaSqabeaacqGHsislcaaIXaaaaOGaeyypa0ZaaeWaaeaafaqa beGacaaabaGaaCiwamaaCaaaleqabaGaamivaaaakiaahkeacaWHyb aabaGaaCiwamaaCaaaleqabaGaamivaaaakiaahkeacaWHAbaabaGa aCOwamaaCaaaleqabaGaamivaaaakiaahkeacaWHybaabaGabC4Ody aajaaaaaGaayjkaiaawMcaamaaCaaaleqabaGaeyOeI0IaaGymaaaa kiabg2da9maabmaabaqbaeaabiGaaaqaaiqahgeagaqcamaaBaaale aacaaIXaGaaGymaaqabaaakeaaceWHbbGbaKaadaWgaaWcbaGaaGym aiaaikdaaeqaaaGcbaGabCyqayaajaWaaSbaaSqaaiaaikdacaaIXa aabeaaaOqaaiqahgeagaqcamaaBaaaleaacaaIYaGaaGOmaaqabaaa aaGccaGLOaGaayzkaaaaaa@5616@

où nous avons partitionné tant V ^ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabCOvayaaja aaaa@36E6@ que son inverse V ^ 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabCOvayaaja WaaWbaaSqabeaacqGHsislcaaIXaaaaaaa@38BB@ selon les dimensions de β MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCOSdaaa@3735@ et u . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCyDaiaac6 caaaa@37A7@ Dans ce cas, A ^ 11 = [ X T B X X T B Z Σ ^ 1 Z T B X ] 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabCyqayaaja WaaSbaaSqaaiaaigdacaaIXaaabeaakiabg2da9maadmaabaGaaCiw amaaCaaaleqabaGaamivaaaakiaahkeacaWHybGaeyOeI0IaaCiwam aaCaaaleqabaGaamivaaaakiaahkeacaWHAbGabC4OdyaajaWaaWba aSqabeaacqGHsislcaaIXaaaaOGaaCOwamaaCaaaleqabaGaamivaa aakiaahkeacaWHybaacaGLBbGaayzxaaWaaWbaaSqabeaacqGHsisl caaIXaaaaaaa@4C30@ et A ^ 22 = Σ ^ 1 + Σ ^ 1 { ( Z T B X ) A ^ 11 ( X T B Z ) } Σ ^ 1 . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabCyqayaaja WaaSbaaSqaaiaaikdacaaIYaaabeaakiabg2da9iqaho6agaqcamaa CaaaleqabaGaeyOeI0IaaGymaaaakiabgUcaRiqaho6agaqcamaaCa aaleqabaGaeyOeI0IaaGymaaaakmaacmaabaWaaeWaaeaacaWHAbWa aWbaaSqabeaacaWGubaaaOGaaCOqaiaahIfaaiaawIcacaGLPaaace WHbbGbaKaadaWgaaWcbaGaaGymaiaaigdaaeqaaOWaaeWaaeaacaWH ybWaaWbaaSqabeaacaWGubaaaOGaaCOqaiaahQfaaiaawIcacaGLPa aaaiaawUhacaGL9baaceWHJoGbaKaadaahaaWcbeqaaiabgkHiTiaa igdaaaGccaGGUaaaaa@537A@ Nous posons Δ = Z * Σ ^ 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuiLdqecbe Gaa8xpaiaahQfadaahaaWcbeqaaiaacQcaaaGcceWHJoGbaKaadaah aaWcbeqaaiabgkHiTiaaigdaaaaaaa@3D01@ ; soit Z ( k ) * MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCOwamaaDa aaleaadaqadaqaaiaadUgaaiaawIcacaGLPaaaaeaacaGGQaaaaaaa @3A2E@ la k e MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4AamaaCa aaleqabaGaaeyzaaaaaaa@37FC@ ligne de la matrice Z * , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCOwamaaCa aaleqabaGaaiOkaaaakiaaygW7caGGSaaaaa@39F9@ dont la dérivée est donnée par

( k ) = Δ ( k ) σ u 2 | σ u 2 = σ ^ u 2 = ( Z ( k ) * Σ 1 ) σ u 2 | σ u 2 = σ ^ u 2 = ( σ ^ u 2 ) 2 Z ( k ) * Σ ^ 1 Σ ^ 1 = Z ( k ) * Σ ^ 1 Ω ^ 1 Ω ^ 1 Σ ^ 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaey4bIe9aaS baaSqaamaabmaabaGaam4AaaGaayjkaiaawMcaaaqabaGccqGH9aqp daabcaqaamaalaaabaGaeyOaIyRaeuiLdq0aaSbaaSqaamaabmaaba Gaam4AaaGaayjkaiaawMcaaaqabaaakeaacqGHciITcqaHdpWCdaqh aaWcbaGaamyDaaqaaiaaikdaaaaaaOGaaGPaVdGaayjcSdWaaSbaaS qaaiabeo8aZnaaDaaameaacaWG1baabaGaaGOmaaaaliabg2da9iqb eo8aZzaajaWaa0baaWqaaiaadwhaaeaacaaIYaaaaaWcbeaakiabg2 da9maaeiaabaWaaSaaaeaacqGHciITdaqadaqaaiaahQfadaqhaaWc baWaaeWaaeaacaWGRbaacaGLOaGaayzkaaaabaGaaiOkaaaakiaaho 6adaahaaWcbeqaaiabgkHiTiaaigdaaaaakiaawIcacaGLPaaaaeaa cqGHciITcqaHdpWCdaqhaaWcbaGaamyDaaqaaiaaikdaaaaaaOGaaG PaVdGaayjcSdWaaSbaaSqaaiabeo8aZnaaDaaameaacaWG1baabaGa aGOmaaaaliabg2da9iqbeo8aZzaajaWaa0baaWqaaiaadwhaaeaaca aIYaaaaaWcbeaakiabg2da9maabmaabaGafq4WdmNbaKaadaqhaaWc baGaamyDaaqaaiaaikdaaaaakiaawIcacaGLPaaadaahaaWcbeqaai abgkHiTiaaikdaaaGccaWHAbWaa0baaSqaamaabmaabaGaam4AaaGa ayjkaiaawMcaaaqaaiaacQcaaaGcceWHJoGbaKaadaahaaWcbeqaai abgkHiTiaaigdaaaGcceWHJoGbaKaadaahaaWcbeqaaiabgkHiTiaa igdaaaGccqGH9aqpcaWHAbWaa0baaSqaamaabmaabaGaam4AaaGaay jkaiaawMcaaaqaaiaacQcaaaGcceWHJoGbaKaadaahaaWcbeqaaiab gkHiTiaaigdaaaGcceWHPoGbaKaadaahaaWcbeqaaiabgkHiTiaaig daaaGcceWHPoGbaKaadaahaaWcbeqaaiabgkHiTiaaigdaaaGcceWH JoGbaKaadaahaaWcbeqaaiabgkHiTiaaigdaaaaaaa@924B@

Σ ^ + = Z T ( B + B Z Ω ^ Z T B ) Z . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabC4Odyaaja WaaWbaaSqabeaacqGHRaWkaaGccqGH9aqpcaWHAbWaaWbaaSqabeaa caWGubaaaOWaaeWaaeaacaWHcbGaey4kaSIaaCOqaiaahQfaceWHPo GbaKaacaWHAbWaaWbaaSqabeaacaWGubaaaOGaaCOqaaGaayjkaiaa wMcaaiaahQfacaGGUaaaaa@45C4@ Dans cette notation, si nous supposons que le modèle (3.2) se vérifie et nous reportons aux tailles et aux chiffres d’échantillon efficaces, les composantes de l’estimation EQM (3.12) sont

m 1 i ( σ ^ u 2 ) = z i T ( Z * Σ ^ 1 Z * T ) z i , m 2 i ( σ ^ u 2 ) = z i T ( C A ^ 11 C T ) z i avec C = X * Z * Σ ^ 1 Z T B X , et m 3 i ( σ ^ u 2 ) = z i T { trace [ ( ( k ) Σ ^ + ( l ) T ) v ( σ ^ u 2 ) ] } z i . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqbaeaabmGaaa qaaiaad2gadaWgaaWcbaGaaGymaiaadMgaaeqaaOWaaeWaaeaacuaH dpWCgaqcamaaDaaaleaacaWG1baabaGaaGOmaaaaaOGaayjkaiaawM caaiabg2da9aqaaiaahQhadaqhaaWcbaGaamyAaaqaaiaadsfaaaGc daqadaqaaiaahQfadaahaaWcbeqaaiaacQcaaaGcceWHJoGbaKaada ahaaWcbeqaaiabgkHiTiaaigdaaaGccaWHAbWaaWbaaSqabeaacaGG QaGaamivaaaaaOGaayjkaiaawMcaaiaahQhadaWgaaWcbaGaamyAaa qabaGccaGGSaaabaGaamyBamaaBaaaleaacaaIYaGaamyAaaqabaGc daqadaqaaiqbeo8aZzaajaWaa0baaSqaaiaadwhaaeaacaaIYaaaaa GccaGLOaGaayzkaaGaeyypa0dabaGaaCOEamaaDaaaleaacaWGPbaa baGaamivaaaakmaabmaabaGaaC4qaiqahgeagaqcamaaBaaaleaaca aIXaGaaGymaaqabaGccaWHdbWaaWbaaSqabeaacaWGubaaaaGccaGL OaGaayzkaaGaaCOEamaaBaaaleaacaWGPbaabeaakiaaysW7caaMc8 UaaeyyaiaabAhacaqGLbGaae4yaiaaysW7caaMc8UaaC4qaGqabiaa =1dacaWHybWaaWbaaSqabeaacaGGQaaaaOGaeyOeI0IaaCOwamaaCa aaleqabaGaaiOkaaaakiqaho6agaqcamaaCaaaleqabaGaeyOeI0Ia aGymaaaakiaahQfadaahaaWcbeqaaiaadsfaaaGccaWHcbGaaCiwai aacYcacaaMe8UaaGPaVlaabwgacaqG0baabaGaamyBamaaBaaaleaa caaIZaGaamyAaaqabaGcdaqadaqaaiqbeo8aZzaajaWaa0baaSqaai aadwhaaeaacaaIYaaaaaGccaGLOaGaayzkaaGaeyypa0dabaGaaCOE amaaDaaaleaacaWGPbaabaGaamivaaaakmaacmaabaGaaeiDaiaabk hacaqGHbGaae4yaiaabwgadaWadaqaamaabmaabaacceGae43bIe9a aSbaaSqaamaabmaabaGaam4AaaGaayjkaiaawMcaaaqabaGcceWHJo GbaKaadaahaaWcbeqaaiabgUcaRaaakiab+DGirpaaDaaaleaadaqa daqaaiaadYgaaiaawIcacaGLPaaaaeaacaWGubaaaaGccaGLOaGaay zkaaGaamODamaabmaabaGafq4WdmNbaKaadaqhaaWcbaGaamyDaaqa aiaaikdaaaaakiaawIcacaGLPaaaaiaawUfacaGLDbaaaiaawUhaca GL9baacaWH6bWaaSbaaSqaaiaadMgaaeqaaOGaaiOlaaaaaaa@AA80@

Là, v ( σ ^ u 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODamaabm aabaGafq4WdmNbaKaadaqhaaWcbaGaamyDaaqaaiaaikdaaaaakiaa wIcacaGLPaaaaaa@3C3B@ est la matrice des covariances asymptotiques de σ ^ u 2 , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafq4WdmNbaK aadaqhaaWcbaGaamyDaaqaaiaaikdaaaGccaGGSaaaaa@3A67@ qui s’obtient comme l’inverse de la matrice d’information de Fisher appropriée pour σ ^ u 2 . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafq4WdmNbaK aadaqhaaWcbaGaamyDaaqaaiaaikdaaaGccaGGUaaaaa@3A69@ Si nous employons l’estimation du maximum de vraisemblance restreint de σ u 2 , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aa0 baaSqaaiaadwhaaeaacaaIYaaaaOGaaiilaaaa@3A57@ v ( σ ^ u 2 ) = 2 ( ( σ ^ u 2 ) 2 ( D 2 t 1 ) + ( σ ^ u 2 ) 4 t 11 ) 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODamaabm aabaGafq4WdmNbaKaadaqhaaWcbaGaamyDaaqaaiaaikdaaaaakiaa wIcacaGLPaaaieqacaWF9aGaaGOmamaabmaabaWaaeWaaeaacuaHdp WCgaqcamaaDaaaleaacaWG1baabaGaaGOmaaaaaOGaayjkaiaawMca amaaCaaaleqabaGaeyOeI0IaaGOmaaaakmaabmaabaGaamiraiabgk HiTiaaikdacaWG0bWaaSbaaSqaaiaaigdaaeqaaaGccaGLOaGaayzk aaGaey4kaSYaaeWaaeaacuaHdpWCgaqcamaaDaaaleaacaWG1baaba GaaGOmaaaaaOGaayjkaiaawMcaamaaCaaaleqabaGaeyOeI0IaaGin aaaakiaadshadaWgaaWcbaGaaGymaiaaigdaaeqaaaGccaGLOaGaay zkaaWaaWbaaSqabeaacqGHsislcaaIXaaaaaaa@58DD@ avec t 1 = ( σ ^ u 2 ) 1 trace ( A ^ 22 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiDamaaBa aaleaacaaIXaaabeaakiabg2da9maabmaabaGafq4WdmNbaKaadaqh aaWcbaGaamyDaaqaaiaaikdaaaaakiaawIcacaGLPaaadaahaaWcbe qaaiabgkHiTiaaigdaaaGccaaMe8UaaeiDaiaabkhacaqGHbGaae4y aiaabwgadaqadaqaaiqahgeagaqcamaaBaaaleaacaaIYaGaaGOmaa qabaaakiaawIcacaGLPaaaaaa@4A4B@ et t 11 = trace ( A ^ 22 A ^ 22 ) . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiDamaaBa aaleaacaaIXaGaaGymaaqabaGccqGH9aqpcaqG0bGaaeOCaiaabgga caqGJbGaaeyzamaabmaabaGabCyqayaajaWaaSbaaSqaaiaaikdaca aIYaaabeaakiqahgeagaqcamaaBaaaleaacaaIYaGaaGOmaaqabaaa kiaawIcacaGLPaaacaGGUaaaaa@458B@

L’estimation EQM du prédicteur empirique spatial PES (3.11) se définit de la même manière. Si nous remplaçons Ω ( σ ^ u 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCyQdmaabm aabaGafq4WdmNbaKaadaqhaaWcbaGaamyDaaqaaiaaikdaaaaakiaa wIcacaGLPaaaaaa@3C75@ par Ω s p ( δ ^ ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCyQdmaaBa aaleaacaWGZbGaamiCaaqabaGcdaqadaqaaiqahs7agaqcaaGaayjk aiaawMcaaaaa@3C28@ et u ^ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabCyDayaaja aaaa@3705@ par v ^ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabCODayaaja aaaa@3706@ en (3.12), elle se formule ainsi :

eqm ( P ^ i ( e ) PES ) = m 1 i s p ( δ ^ ) + m 2 i s p ( δ ^ ) + 2 m 3 i s p ( δ ^ ) , ( 3.13 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeyzaiaabg hacaqGTbWaaeWaaeaaceWGqbGbaKaadaqhaaWcbaGaamyAamaabmaa baGaamyzaaGaayjkaiaawMcaaaqaaiaabcfacaqGfbGaae4uaaaaaO GaayjkaiaawMcaaiabg2da9iaad2gadaqhaaWcbaGaaGymaiaadMga aeaacaWGZbGaamiCaaaakmaabmaabaGabCiTdyaajaaacaGLOaGaay zkaaGaey4kaSIaamyBamaaDaaaleaacaaIYaGaamyAaaqaaiaadoha caWGWbaaaOWaaeWaaeaaceWH0oGbaKaaaiaawIcacaGLPaaacqGHRa WkcaaIYaGaamyBamaaDaaaleaacaaIZaGaamyAaaqaaiaadohacaWG WbaaaOWaaeWaaeaaceWH0oGbaKaaaiaawIcacaGLPaaacaGGSaGaaG zbVlaaywW7caaMf8UaaGzbVlaaywW7caGGOaGaaG4maiaac6cacaaI XaGaaG4maiaacMcaaaa@6840@

où dans (3.5) les trois composantes de (3.13) se définissent comme ci-après. Posons

Σ ^ s p = Z T B s p Z + Ω ^ s p 1 , Z s p * = { diag ( N i 1 ) ; i = 1 , , D } H s p Z , X s p * = { diag ( N i 1 ) ; i = 1 , , D } H s p X , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqbaeaabmGaaa qaaiqaho6agaqcamaaBaaaleaacaWGZbGaamiCaaqabaaakeaacqGH 9aqpcaWHAbWaaWbaaSqabeaacaWGubaaaOGaaCOqamaaBaaaleaaca WGZbGaamiCaaqabaGccaWHAbacbeGaa83kaiqahM6agaqcamaaDaaa leaacaWGZbGaamiCaaqaaiabgkHiTiaaigdaaaGccaGGSaaabaGaaC OwamaaDaaaleaacaWGZbGaamiCaaqaaiaacQcaaaaakeaacqGH9aqp daGadaqaaiaabsgacaqGPbGaaeyyaiaabEgadaqadaqaaiaad6eada qhaaWcbaGaamyAaaqaaiabgkHiTiaaigdaaaaakiaawIcacaGLPaaa caGG7aGaaGjbVlaaykW7caWGPbGaeyypa0JaaGymaiaacYcacaaMe8 UaeSOjGSKaaiilaiaaysW7caWGebaacaGL7bGaayzFaaGaaCisamaa BaaaleaacaWGZbGaamiCaaqabaGccaWHAbGaaiilaaqaaiaahIfada qhaaWcbaGaam4CaiaadchaaeaacaGGQaaaaaGcbaGaeyypa0ZaaiWa aeaacaqGKbGaaeyAaiaabggacaqGNbWaaeWaaeaacaWGobWaa0baaS qaaiaadMgaaeaacqGHsislcaaIXaaaaaGccaGLOaGaayzkaaGaai4o aiaaysW7caaMc8UaamyAaiabg2da9iaaigdacaGGSaGaaGjbVlablA ciljaacYcacaaMe8UaamiraaGaay5Eaiaaw2haaiaahIeadaWgaaWc baGaam4CaiaadchaaeqaaOGaaCiwaiaacYcaaaaaaa@88F8@

avec

B s p = diag { n d P ^ i ( e ) PES ( 1 P ^ i ( e ) PES ) ; i = 1 , , D } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCOqamaaBa aaleaacaWGZbGaamiCaaqabaGccqGH9aqpcaqGKbGaaeyAaiaabgga caqGNbWaaiWaaeaacaWGUbWaaSbaaSqaaiaadsgaaeqaaOGabmiuay aajaWaa0baaSqaaiaadMgadaqadaqaaiaadwgaaiaawIcacaGLPaaa aeaacaqGqbGaaeyraiaabofaaaGcdaqadaqaaiaaigdacqGHsislce WGqbGbaKaadaqhaaWcbaGaamyAamaabmaabaGaamyzaaGaayjkaiaa wMcaaaqaaiaabcfacaqGfbGaae4uaaaaaOGaayjkaiaawMcaaiaacU dacaaMe8UaaGPaVlaadMgacqGH9aqpcaaIXaGaaiilaiaaysW7cqWI MaYscaGGSaGaaGjbVlaadseaaiaawUhacaGL9baaaaa@5FC6@

et

H s p = diag { P ^ i ( e ) PES ( 1 P ^ i ( e ) PES ) ; i = 1 , , D } . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCisamaaBa aaleaacaWGZbGaamiCaaqabaGccqGH9aqpcaqGKbGaaeyAaiaabgga caqGNbWaaiWaaeaaceWGqbGbaKaadaqhaaWcbaGaamyAamaabmaaba GaamyzaaGaayjkaiaawMcaaaqaaiaabcfacaqGfbGaae4uaaaakmaa bmaabaGaaGymaiabgkHiTiqadcfagaqcamaaDaaaleaacaWGPbWaae WaaeaacaWGLbaacaGLOaGaayzkaaaabaGaaeiuaiaabweacaqGtbaa aaGccaGLOaGaayzkaaGaai4oaiaaysW7caaMc8UaamyAaiabg2da9i aaigdacaGGSaGaaGjbVlablAciljaacYcacaaMe8UaamiraaGaay5E aiaaw2haaiaac6caaaa@5E6C@

Alors

m 1 i s p ( δ ^ ) = z i T ( Z s p * Σ ^ s p 1 Z s p * T ) z i , m 2 i s p ( δ ^ ) = z i T { C s p ( X T B s p X X T B s p Z Σ ^ s p 1 Z T B s p X ) 1 C s p T } z i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqbaeaabiGaaa qaaiaad2gadaqhaaWcbaGaaGymaiaadMgaaeaacaWGZbGaamiCaaaa kmaabmaabaGabCiTdyaajaaacaGLOaGaayzkaaaabaGaeyypa0JaaC OEamaaDaaaleaacaWGPbaabaGaamivaaaakmaabmaabaGaaCOwamaa DaaaleaacaWGZbGaamiCaaqaaiaacQcaaaGcceWHJoGbaKaadaqhaa WcbaGaam4CaiaadchaaeaacqGHsislcaaIXaaaaOGaaCOwamaaDaaa leaacaWGZbGaamiCaaqaaiaacQcacaWGubaaaaGccaGLOaGaayzkaa GaaCOEamaaBaaaleaacaWGPbaabeaakiaacYcaaeaacaWGTbWaa0ba aSqaaiaaikdacaWGPbaabaGaam4CaiaadchaaaGcdaqadaqaaiqahs 7agaqcaaGaayjkaiaawMcaaaqaaiabg2da9iaahQhadaqhaaWcbaGa amyAaaqaaiaadsfaaaGcdaGadaqaaiaahoeadaWgaaWcbaGaam4Cai aadchaaeqaaOWaaeWaaeaacaWHybWaaWbaaSqabeaacaWGubaaaOGa aCOqamaaBaaaleaacaWGZbGaamiCaaqabaGccaWHybGaeyOeI0IaaC iwamaaCaaaleqabaGaamivaaaakiaahkeadaWgaaWcbaGaam4Caiaa dchaaeqaaOGaaCOwaiqaho6agaqcamaaDaaaleaacaWGZbGaamiCaa qaaiabgkHiTiaaigdaaaGccaWHAbWaaWbaaSqabeaacaWGubaaaOGa aCOqamaaBaaaleaacaWGZbGaamiCaaqabaGccaWHybaacaGLOaGaay zkaaWaaWbaaSqabeaacqGHsislcaaIXaaaaOGaaC4qamaaDaaaleaa caWGZbGaamiCaaqaaiaadsfaaaaakiaawUhacaGL9baacaWH6bWaaS baaSqaaiaadMgaaeqaaaaaaaa@849C@

avec C s p = X s p * Z s p * Σ ^ s p 1 Z T B s p X , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaC4qamaaBa aaleaacaWGZbGaamiCaaqabaGccqGH9aqpcaWHybWaa0baaSqaaiaa dohacaWGWbaabaGaaiOkaaaakiabgkHiTiaahQfadaqhaaWcbaGaam 4CaiaadchaaeaacaGGQaaaaOGabC4OdyaajaWaa0baaSqaaiaadoha caWGWbaabaGaeyOeI0IaaGymaaaakiaahQfadaahaaWcbeqaaiaads faaaGccaWHcbWaaSbaaSqaaiaadohacaWGWbaabeaakiaahIfacaGG Saaaaa@4DBE@ et

m 3 i s p ( δ ^ ) = z i T { trace [ ( ( k ) s p Σ ^ s p + ( l ) s p T ) v ( δ ^ ) ] } z i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBamaaDa aaleaacaaIZaGaamyAaaqaaiaadohacaWGWbaaaOWaaeWaaeaaceWH 0oGbaKaaaiaawIcacaGLPaaacqGH9aqpcaWH6bWaa0baaSqaaiaadM gaaeaacaWGubaaaOWaaiWaaeaacaqG0bGaaeOCaiaabggacaqGJbGa aeyzamaadmaabaWaaeWaaeaaiiqacqWFhis0daqhaaWcbaWaaeWaae aacaWGRbaacaGLOaGaayzkaaaabaGaam4CaiaadchaaaGcceWHJoGb aKaadaqhaaWcbaGaam4CaiaadchaaeaacqGHRaWkaaGccqWFhis0da qhaaWcbaWaaeWaaeaacaWGSbaacaGLOaGaayzkaaaabaGaam4Caiaa dchacaWGubaaaaGccaGLOaGaayzkaaGaamODamaabmaabaGabCiTdy aajaaacaGLOaGaayzkaaaacaGLBbGaayzxaaaacaGL7bGaayzFaaGa aCOEamaaBaaaleaacaWGPbaabeaaaaa@6334@

avec Σ ^ s p + = Z T ( B s p + B s p Z Ω ^ s p Z T B s p ) Z . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabC4Odyaaja Waa0baaSqaaiaadohacaWGWbaabaGaey4kaScaaOGaeyypa0JaaCOw amaaCaaaleqabaGaamivaaaakmaabmaabaGaaCOqamaaBaaaleaaca WGZbGaamiCaaqabaGccqGHRaWkcaWHcbWaaSbaaSqaaiaadohacaWG WbaabeaakiaahQfaceWHPoGbaKaadaWgaaWcbaGaam4Caiaadchaae qaaOGaaCOwamaaCaaaleqabaGaamivaaaakiaahkeadaWgaaWcbaGa am4CaiaadchaaeqaaaGccaGLOaGaayzkaaGaaCOwaiaac6caaaa@503D@ Ici, v ( δ ^ ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODamaabm aabaGabCiTdyaajaaacaGLOaGaayzkaaaaaa@39CB@ est la matrice des covariances asymptotiques des estimateurs des paramètres δ ^ , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabCiTdyaaja Gaaiilaaaa@37F7@ des composantes de variance

( k ) s p = Δ ( k ) s p δ ^ | δ ^ = δ = ( Z s p ( k ) * Σ s p 1 ) ( σ u 2 , ρ ) T | δ ^ = δ = Z s p ( k ) * Σ ^ s p 1 Ω s p 1 ( δ ^ ) Ω s p 1 ( δ ^ ) Σ ^ s p 1 , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacceGae83bIe 9aa0baaSqaamaabmaabaGaam4AaaGaayjkaiaawMcaaaqaaiaadoha caWGWbaaaOGaeyypa0ZaaqGaaeaadaWcaaqaaiabgkGi2kabfs5aen aaDaaaleaadaqadaqaaiaadUgaaiaawIcacaGLPaaaaeaacaWGZbGa amiCaaaaaOqaaiabgkGi2kqahs7agaqcaaaacaaMc8oacaGLiWoada WgaaWcbaGabCiTdyaajaGaeyypa0JaaCiTdaqabaGccqGH9aqpdaab caqaamaalaaabaGaeyOaIy7aaeWaaeaacaWHAbWaa0baaSqaaGqaci aa+jcacaWGZbGaamiCamaabmaabaGaam4AaaGaayjkaiaawMcaaaqa aiaacQcaaaGccaWHJoWaa0baaSqaaiaadohacaWGWbaabaGaeyOeI0 IaaGymaaaaaOGaayjkaiaawMcaaaqaaiabgkGi2oaabmaabaGaeq4W dm3aa0baaSqaaiaadwhaaeaacaaIYaaaaOGaaiilaiaaysW7cqaHbp GCaiaawIcacaGLPaaadaahaaWcbeqaaiaadsfaaaaaaOGaaGPaVdGa ayjcSdWaaSbaaSqaaiqahs7agaqcaiabg2da9iaahs7aaeqaaOGaey ypa0JaaCOwamaaDaaaleaacaWGZbGaamiCamaabmaabaGaam4AaaGa ayjkaiaawMcaaaqaaiaacQcaaaGcceWHJoGbaKaadaqhaaWcbaGaam 4CaiaadchaaeaacqGHsislcaaIXaaaaOGaaCyQdmaaDaaaleaacaWG ZbGaamiCaaqaaiabgkHiTiaaigdaaaGcdaqadaqaaiqahs7agaqcaa GaayjkaiaawMcaaiaahM6adaqhaaWcbaGaam4CaiaadchaaeaacqGH sislcaaIXaaaaOWaaeWaaeaaceWH0oGbaKaaaiaawIcacaGLPaaace WHJoGbaKaadaqhaaWcbaGaam4CaiaadchaaeaacqGHsislcaaIXaaa aOGaaiilaaaa@9261@

Δ s p = Z s p * Σ ^ s p 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuiLdq0aaW baaSqabeaacaWGZbGaamiCaaaaieqakiaa=1dacaWHAbWaa0baaSqa aiaadohacaWGWbaabaGaaiOkaaaakiqaho6agaqcamaaDaaaleaaca WGZbGaamiCaaqaaiabgkHiTiaaigdaaaaaaa@42FF@ and Z s p ( k ) * MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCOwamaaDa aaleaacaWGZbGaamiCamaabmaabaGaam4AaaGaayjkaiaawMcaaaqa aiaacQcaaaaaaa@3C1B@ est la k e MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4AamaaCa aaleqabaGaaeyzaaaaaaa@37FC@ ligne de Z s p * . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCOwamaaDa aaleaacaWGZbGaamiCaaqaaiaacQcaaaGccaGGUaaaaa@3A5E@


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