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

Section 4. Modèle au niveau de l’unité

Le modèle original au niveau de l’unité a été proposé par Battese et coll. (1988). Ces derniers ont supposé le modèle à erreurs emboîtées suivant :

y i j = z i j T β + v i + e i j pour i = 1 , , m et j U i ( 4.1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEamaaBa aaleaacaWGPbGaamOAaaqabaGccqGH9aqpcaWH6bWaa0baaSqaaiaa dMgacaWGQbaabaGaamivaaaakiaahk7acqGHRaWkcaWG2bWaaSbaaS qaaiaadMgaaeqaaOGaey4kaSIaamyzamaaBaaaleaacaWGPbGaamOA aaqabaGccaaMf8UaaeiCaiaab+gacaqG1bGaaeOCaiaaywW7caWGPb Gaeyypa0JaaGymaiaacYcacaaMe8UaeSOjGSKaaiilaiaaysW7caWG TbGaaGzbVlaabwgacaqG0bGaaGzbVlaadQgacqGHiiIZcaWGvbWaaS baaSqaaiaadMgaaeqaaOGaaGzbVlaaywW7caaMf8UaaGzbVlaaywW7 caGGOaGaaGinaiaac6cacaaIXaGaaiykaaaa@6AFC@

v i ind ( 0 , σ v 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODamaaBa aaleaacaWGPbaabeaakiaaysW7daGfGbqabSqabeaacaqGPbGaaeOB aiaabsgaaeaarqqr1ngBPrgifHhDYfgaiuaajugybiab=XJi6aaaki aaysW7daqadaqaaiaaicdacaGGSaGaaGjbVlabeo8aZnaaDaaaleaa caWG2baabaGaaGOmaaaaaOGaayjkaiaawMcaaaaa@4D41@ sont les effets aléatoires et sont indépendants des erreurs aléatoires, e i j , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyzamaaBa aaleaacaWGPbGaaGzaVlaadQgaaeqaaOGaaiilaaaa@3B2E@ avec e i j ind ( 0 , σ e 2 ) . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyzamaaBa aaleaacaWGPbGaaGzaVlaadQgaaeqaaOGaaGjbVpaawagabeWcbeqa aiaabMgacaqGUbGaaeizaaqaaebbfv3ySLgzGueE0jxyaGqbaKqzGf Gae8hpIOdaaOGaaGjbVpaabmaabaGaaGimaiaacYcacaaMe8Uaeq4W dm3aa0baaSqaaiaadwgaaeaacaaIYaaaaaGccaGLOaGaayzkaaGaai Olaaaa@504A@ Le système de production comprend une légère modification de la structure des erreurs aléatoires, c’est-à-dire que e i j ind ( 0 , σ e 2 / a i j ) , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyzamaaBa aaleaacaWGPbGaamOAaaqabaGccaaMe8+aaybyaeqaleqabaGaaeyA aiaab6gacaqGKbaabaqeeuuDJXwAKbsr4rNCHbacfaqcLbwacqWF8i IoaaGccaaMe8+aaeWaaeaacaaIWaGaaiilaiaaysW7daWcgaqaaiab eo8aZnaaDaaaleaacaWGLbaabaGaaGOmaaaaaOqaaiaadggadaWgaa WcbaGaamyAaiaadQgaaeqaaaaaaOGaayjkaiaawMcaaiaacYcaaaa@51CD@ a i j > 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaBa aaleaacaWGPbGaaGzaVlaadQgaaeqaaOGaeyOpa4JaaGimaaaa@3C3C@ sont des constantes positives qui rendent compte de l’hétéroscédasticité.

Le système de production calcule les estimations sur petits domaines pour les moyennes ( Y ¯ i c = j U i c i j y i j / j U i c i j ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaace WGzbGbaebadaWgaaWcbaGaamyAaiaadogaaeqaaOGaeyypa0ZaaSGb aeaadaaeqaqaaiaadogadaWgaaWcbaGaamyAaiaadQgaaeqaaOGaam yEamaaBaaaleaacaWGPbGaamOAaaqabaaabaGaamOAaiabgIGiolaa dwfadaWgaaadbaGaamyAaaqabaaaleqaniabggHiLdaakeaadaaeqa qaaiaadogadaWgaaWcbaGaamyAaiaadQgaaeqaaaqaaiaadQgacqGH iiIZcaWGvbWaaSbaaWqaaiaadMgaaeqaaaWcbeqdcqGHris5aaaaaO GaayjkaiaawMcaaaaa@513B@ et les totaux ( Y i c = j U i c i j Y ¯ i ) . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaaca WGzbWaaSbaaSqaaiaadMgacaWGJbaabeaakiabg2da9maaqababaGa am4yamaaBaaaleaacaWGPbGaamOAaaqabaGcceWGzbGbaebadaWgaa WcbaGaamyAaaqabaaabaGaamOAaiabgIGiolaadwfadaWgaaadbaGa amyAaaqabaaaleqaniabggHiLdaakiaawIcacaGLPaaacaGGUaaaaa@4782@ Les valeurs c i j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4yamaaBa aaleaacaWGPbGaamOAaaqabaaaaa@38E8@ sont des constantes positives fixes connues pour toutes les unités de la population. Il a fallu ajouter c i j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4yamaaBa aaleaacaWGPbGaamOAaaqabaaaaa@38E8@ afin que certaines enquêtes-entreprises menées à Statistique Canada puissent utiliser le système (voir Rubin-Bleuer, Jang et Godbout, 2016). Les données auxiliaires disponibles sont des totaux Z i c = j U i c i j z i j , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCOwamaaBa aaleaacaWGPbGaam4yaaqabaGccqGH9aqpdaaeqaqaaiaadogadaWg aaWcbaGaamyAaiaadQgaaeqaaOGaaCOEamaaBaaaleaacaWGPbGaam OAaaqabaaabaGaamOAaiabgIGiolaadwfadaWgaaadbaGaamyAaaqa baaaleqaniabggHiLdGccaGGSaaaaa@46F8@ ou des moyennes Z ¯ i c = j U i c i j z i j / j U i c i j . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabCOwayaara WaaSbaaSqaaiaadMgacaWGJbaabeaakiabg2da9maalyaabaWaaabe aeaacaWGJbWaaSbaaSqaaiaadMgacaWGQbaabeaakiaahQhadaWgaa WcbaGaamyAaiaadQgaaeqaaaqaaiaadQgacqGHiiIZcaWGvbWaaSba aWqaaiaadMgaaeqaaaWcbeqdcqGHris5aaGcbaWaaabeaeaacaWGJb WaaSbaaSqaaiaadMgacaWGQbaabeaaaeaacaWGQbGaeyicI4Saamyv amaaBaaameaacaWGPbaabeaaaSqab0GaeyyeIuoaaaGccaGGUaaaaa@506E@

Dans ce qui suit, nous fournissons les estimateurs de la moyenne de population Y ¯ i c , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmywayaara WaaSbaaSqaaiaadMgacaWGJbaabeaakiaacYcaaaa@39A9@ disons θ ^ i EPD , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiUdeNbaK aadaqhaaWcbaGaamyAaaqaaiaabweacaqGqbGaaeiraaaakiaacYca aaa@3BF4@ i = 1 , , M . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAaiabg2 da9iaaigdacaGGSaGaaGjbVlablAciljaacYcacaaMe8Uaamytaiaa c6caaaa@3FC6@ Les estimations des totaux correspondants Y i c , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamywamaaBa aaleaacaWGPbGaam4yaaqabaGccaGGSaaaaa@3991@ sont obtenues en multipliant θ ^ i EPD MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiUdeNbaK aadaqhaaWcbaGaamyAaaqaaiaabweacaqGqbGaaeiraaaaaaa@3B3A@ par j = 1 N i c i j . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaabmaeaaca WGJbWaaSbaaSqaaiaadMgacaWGQbaabeaaaeaacaWGQbGaeyypa0Ja aGymaaqaaiaad6eadaWgaaadbaGaamyAaaqabaaaniabggHiLdGcca GGUaaaaa@4039@

La moyenne de l’échantillon pondéré déterminé par le plan d’échantillonnage des y MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEaaaa@36F5@ et des z MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCOEaaaa@36FA@ est respectivement :

y ¯ i w c = ( j s i w i j c i j ) 1 j s i w i j c i j y i j MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmyEayaara WaaSbaaSqaaiaadMgacGaDao4DaiaadogaaeqaaOGaeyypa0ZaaeWa beaadaaeqbqaaiaadEhadaWgaaWcbaGaamyAaiaadQgaaeqaaOGaam 4yamaaBaaaleaacaWGPbGaamOAaaqabaaabaGaamOAaiabgIGiolaa dohadaWgaaadbaGaamyAaaqabaaaleqaniabggHiLdaakiaawIcaca GLPaaadaahaaWcbeqaaiabgkHiTiaaigdaaaGcdaaeqbqaaiaadEha daWgaaWcbaGaamyAaiaadQgaaeqaaOGaam4yamaaBaaaleaacaWGPb GaamOAaaqabaGccaWG5bWaaSbaaSqaaiaadMgacaWGQbaabeaaaeaa caWGQbGaeyicI4Saam4CamaaBaaameaacaWGPbaabeaaaSqab0Gaey yeIuoaaaa@5BEC@

et

z ¯ i w c = ( j s i w i j c i j ) 1 j s i w i j c i j z i j . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabCOEayaara WaaSbaaSqaaiaadMgacaWG3bGaam4yaaqabaGccqGH9aqpdaqadeqa amaaqafabaGaam4DamaaBaaaleaacaWGPbGaamOAaaqabaGccaWGJb WaaSbaaSqaaiaadMgacaWGQbaabeaaaeaacaWGQbGaeyicI4Saam4C amaaBaaameaacaWGPbaabeaaaSqab0GaeyyeIuoaaOGaayjkaiaawM caamaaCaaaleqabaGaeyOeI0IaaGymaaaakmaaqafabaGaam4Damaa BaaaleaacaWGPbGaamOAaaqabaGccaWGJbWaaSbaaSqaaiaadMgaca WGQbaabeaakiaahQhadaWgaaWcbaGaamyAaiaadQgaaeqaaaqaaiaa dQgacqGHiiIZcaWGZbWaaSbaaWqaaiaadMgaaeqaaaWcbeqdcqGHri s5aOGaaiOlaaaa@5BB6@

Les moyennes pondérées fondées sur le modèle sont les suivantes :

y ¯ i a = ( j s i a i j ) 1 ( j s i a i j y i j ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmyEayaara WaaSbaaSqaaiaadMgacaWGHbaabeaakiabg2da9maabmqabaWaaabu aeaacaWGHbWaaSbaaSqaaiaadMgacaWGQbaabeaaaeaacaWGQbGaey icI4Saam4CamaaBaaameaacaWGPbaabeaaaSqab0GaeyyeIuoaaOGa ayjkaiaawMcaamaaCaaaleqabaGaeyOeI0IaaGymaaaakmaabmqaba WaaabuaeaacaWGHbWaaSbaaSqaaiaadMgacaWGQbaabeaakiaadMha daWgaaWcbaGaamyAaiaadQgaaeqaaaqaaiaadQgacqGHiiIZcaWGZb WaaSbaaWqaaiaadMgaaeqaaaWcbeqdcqGHris5aaGccaGLOaGaayzk aaaaaa@5564@

et

z ¯ i a = ( j s i a i j ) 1 ( j s i a i j z i j ) . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabCOEayaara WaaSbaaSqaaiaadMgacaWGHbaabeaakiabg2da9maabmqabaWaaabu aeaacaWGHbWaaSbaaSqaaiaadMgacaWGQbaabeaaaeaacaWGQbGaey icI4Saam4CamaaBaaameaacaWGPbaabeaaaSqab0GaeyyeIuoaaOGa ayjkaiaawMcaamaaCaaaleqabaGaeyOeI0IaaGymaaaakmaabmqaba WaaabuaeaacaWGHbWaaSbaaSqaaiaadMgacaWGQbaabeaakiaahQha daWgaaWcbaGaamyAaiaadQgaaeqaaaqaaiaadQgacqGHiiIZcaWGZb WaaSbaaWqaaiaadMgaaeqaaaWcbeqdcqGHris5aaGccaGLOaGaayzk aaGaaiOlaaaa@5620@

Battese et coll. (1988) n’ont pas inclus les poids déterminés par le plan d’échantillonnage dans leur procédure, forçant ainsi l’uniformité du plan, à moins qu’il ne soit autopondéré. Nous appelons cet estimateur EBLUP ( θ ^ i EBLUP ) . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaacu aH4oqCgaqcamaaDaaaleaacaWGPbaabaGaaeyraiaabkeacaqGmbGa aeyvaiaabcfaaaaakiaawIcacaGLPaaacaGGUaaaaa@3F24@ Cependant, EBLUP est l’estimateur le plus efficace selon le modèle (4.1), avec une structure d’erreur e i j ind ( 0 , σ e 2 / a i j ) , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyzamaaBa aaleaacaWGPbGaamOAaaqabaGccaaMe8+aaybyaeqaleqabaGaaeyA aiaab6gacaqGKbaabaqeeuuDJXwAKbsr4rNCHbacfaqcLbwacqWF8i IoaaGccaaMe8+aaeWaaeaacaaIWaGaaiilaiaaysW7daWcgaqaaiab eo8aZnaaDaaaleaacaWGLbaabaGaaGOmaaaaaOqaaiaadggadaWgaa WcbaGaamyAaiaaygW7caWGQbaabeaaaaaakiaawIcacaGLPaaacaGG Saaaaa@5357@ et c’est la raison pour laquelle il est inclus dans le système de production.

Kott (1989), Prasad et Rao (1999), et You et Rao (2002) ont proposé d’utiliser des estimateurs fondés sur un modèle et convergents par rapport au plan pour les moyennes des domaines en incluant le poids d’enquête. La procedure de You et Rao (2002) a été modifiée de façon appropriée pour représenter les résidus hétéroscédastiques c i j . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4yamaaBa aaleaacaWGPbGaamOAaaqabaacbaGccaWFUaaaaa@39AA@ Le pseudo-estimateur EBLUP qui en découle, désigné sous le nom de PEBLUP ( θ ^ i PEBLUP ) , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaacu aH4oqCgaqcamaaDaaaleaacaWGPbaabaGaaeiuaiaabweacaqGcbGa aeitaiaabwfacaqGqbaaaaGccaGLOaGaayzkaaGaaiilaaaa@3FF5@ a été inclus dans le système de production parce qu’il est convergent par rapport au plan.

L’estimateur EBLUP est défini comme suit :

θ ^ i EBLUP = { γ ^ i a y ¯ i a + ( Z ¯ i c γ ^ i a z ¯ i a ) T β ^ EBLUP si i A Z ¯ i c T β ^ EBLUP si i A ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiUdeNbaK aadaqhaaWcbaGaamyAaaqaaiaabweacaqGcbGaaeitaiaabwfacaqG qbaaaOGaeyypa0ZaaiqaaeaafaqaaeGacaaabaGafq4SdCMbaKaada WgaaWcbaGaamyAaiaadggaaeqaaOGabmyEayaaraWaaSbaaSqaaiaa dMgacaWGHbaabeaakiabgUcaRmaabmaabaGabCOwayaaraWaaSbaaS qaaiaadMgacaWGJbaabeaakiabgkHiTiqbeo7aNzaajaWaaSbaaSqa aiaadMgacaWGHbaabeaakiqahQhagaqeamaaBaaaleaacaWGPbGaam yyaaqabaaakiaawIcacaGLPaaadaahaaWcbeqaaiaadsfaaaGcceWH YoGbaKaadaahaaWcbeqaaiaabweacaqGcbGaaeitaiaabwfacaqGqb aaaaGcbaGaae4CaiaabMgacaaMe8UaaGPaVlaadMgacqGHiiIZcaWG bbaabaGabCOwayaaraWaa0baaSqaaiaadMgacaWGJbaabaGaamivaa aakiqahk7agaqcamaaCaaaleqabaGaaeyraiaabkeacaqGmbGaaeyv aiaabcfaaaaakeaacaqGZbGaaeyAaiaaysW7caaMc8UaamyAaiabgI GiolqadgeagaqeaaaaaiaawUhaaaaa@73C2@

γ ^ i a = ( σ ^ v 2 + σ ^ e 2 / j s i a i j ) 1 σ ^ v 2 . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafq4SdCMbaK aadaWgaaWcbaGaamyAaiaadggaaeqaaOGaeyypa0ZaaeWaaeaadaWc gaqaaiqbeo8aZzaajaWaa0baaSqaaiaadAhaaeaacaaIYaaaaOGaey 4kaSIafq4WdmNbaKaadaqhaaWcbaGaamyzaaqaaiaaikdaaaaakeaa daaeqaqaaiaadggadaWgaaWcbaGaamyAaiaadQgaaeqaaaqaaiaadQ gacqGHiiIZcaWGZbWaaSbaaWqaaiaadMgaaeqaaaWcbeqdcqGHris5 aaaaaOGaayjkaiaawMcaamaaCaaaleqabaGaeyOeI0IaaGymaaaaki qbeo8aZzaajaWaa0baaSqaaiaadAhaaeaacaaIYaaaaOGaaiOlaaaa @5464@ Les termes y ¯ i a MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmyEayaara WaaSbaaSqaaiaadMgacaWGHbaabeaaaaa@390D@ et z ¯ i a , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabCOEayaara WaaSbaaSqaaiaadMgacaWGHbaabeaakiaacYcaaaa@39CC@ sont les moyennes pondérées fondées sur le modèle déjà définies, respectivement pour y MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEaaaa@36F5@ et z . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCOEaiaac6 caaaa@37AC@ Le vecteur de régression β MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCOSdaaa@3735@ est estimé comme ceci :

β ^ EBLUP = ( i = 1 m j s i a i j c i j ( z i j γ ^ i a c z ¯ i a c ) z i j T ) 1 i = 1 m j s i a i j c i j ( z i j γ ^ i a c z ¯ i a c ) y i j . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabCOSdyaaja WaaWbaaSqabeaacaqGfbGaaeOqaiaabYeacaqGvbGaaeiuaaaakiab g2da9maabmaabaWaaabCaeaadaaeqbqaaiaadggadaWgaaWcbaGaam yAaiaadQgaaeqaaOGaam4yamaaBaaaleaacaWGPbGaamOAaaqabaGc daqadaqaaiaahQhadaWgaaWcbaGaamyAaiaadQgaaeqaaOGaeyOeI0 Iafq4SdCMbaKaadaWgaaWcbaGaamyAaiaadggacaWGJbaabeaakiqa hQhagaqeamaaBaaaleaacaWGPbGaamyyaiaadogaaeqaaaGccaGLOa GaayzkaaGaaCOEamaaDaaaleaacaWGPbGaamOAaaqaaiaadsfaaaaa baGaamOAaiabgIGiolaadohadaWgaaadbaGaamyAaaqabaaaleqani abggHiLdaaleaacaWGPbGaeyypa0JaaGymaaqaaiaad2gaa0Gaeyye IuoaaOGaayjkaiaawMcaamaaCaaaleqabaGaeyOeI0IaaGymaaaakm aaqahabaWaaabuaeaacaWGHbWaaSbaaSqaaiaadMgacaWGQbaabeaa kiaadogadaWgaaWcbaGaamyAaiaadQgaaeqaaOWaaeWaaeaacaWH6b WaaSbaaSqaaiaadMgacaWGQbaabeaakiabgkHiTiqbeo7aNzaajaWa aSbaaSqaaiaadMgacaWGHbGaam4yaaqabaGcceWH6bGbaebadaWgaa WcbaGaamyAaiaadggacaWGJbaabeaaaOGaayjkaiaawMcaaiaadMha daWgaaWcbaGaamyAaiaadQgaaeqaaaqaaiaadQgacqGHiiIZcaWGZb WaaSbaaWqaaiaadMgaaeqaaaWcbeqdcqGHris5aaWcbaGaamyAaiab g2da9iaaigdaaeaacaWGTbaaniabggHiLdGccaGGUaaaaa@8927@

L’estimateur PEBLUP, θ ^ i PEBLUP , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiUdeNbaK aadaqhaaWcbaGaamyAaaqaaiaabcfacaqGfbGaaeOqaiaabYeacaqG vbGaaeiuaaaakiaacYcaaaa@3E6C@ est obtenu comme ceci :

θ ^ i PEBLUP = { γ ^ i w c y ¯ i w c + ( Z ¯ i c γ ^ i w c z ¯ i w c ) T β ^ PEBLUP si i A Z ¯ i c T β ^ PEBLUP si i A ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiUdeNbaK aadaqhaaWcbaGaamyAaaqaaiaabcfacaqGfbGaaeOqaiaabYeacaqG vbGaaeiuaaaakiabg2da9maaceaabaqbaeaabiGaaaqaaiqbeo7aNz aajaWaaSbaaSqaaiaadMgacaWG3bGaam4yaaqabaGcceWG5bGbaeba daWgaaWcbaGaamyAaiac0b4G3bGaam4yaaqabaGccqGHRaWkdaqada qaaiqahQfagaqeamaaBaaaleaacaWGPbGaam4yaaqabaGccqGHsisl c0alas4SdCMbiWcGjaWaiWcGBaaaleacSaOaiWcGdMgacGalao4Dai acSa4GJbaabKalacGcceWH6bGbaebadaWgaaWcbaGaamyAaiaadEha caWGJbaabeaaaOGaayjkaiaawMcaamaaCaaaleqabaGaamivaaaaki qahk7agaqcamaaCaaaleqabaGaaeiuaiaabweacaqGcbGaaeitaiaa bwfacaqGqbaaaaGcbaGaae4CaiaabMgacaaMe8UaaGPaVlaadMgacq GHiiIZcaWGbbaabaGabCOwayaaraWaa0baaSqaaiaadMgacaWGJbaa baGaamivaaaakiqahk7agaqcamaaCaaaleqabaGaaeiuaiaabweaca qGcbGaaeitaiaabwfacaqGqbaaaaGcbaGaae4CaiaabMgacaaMe8Ua aGPaVlaadMgacqGHiiIZceWGbbGbaebaaaaacaGL7baaaaa@80CF@

γ ^ i w c = ( σ ^ v 2 + σ ^ e 2 δ i w c 2 ) 1 ( σ ^ v 2 ) , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafq4SdCMbaK aadaWgaaWcbaGaamyAaiaadEhacaWGJbaabeaakiabg2da9maabmaa baGafq4WdmNbaKaadaqhaaWcbaGaamODaaqaaiaaikdaaaGccqGHRa WkcuaHdpWCgaqcamaaDaaaleaacaWGLbaabaGaaGOmaaaakiabes7a KnaaDaaaleaacaWGPbGaam4DaiaadogaaeaacaaIYaaaaaGccaGLOa GaayzkaaWaaWbaaSqabeaacqGHsislcaaIXaaaaOWaaeWaaeaacuaH dpWCgaqcamaaDaaaleaacaWG2baabaGaaGOmaaaaaOGaayjkaiaawM caaiaacYcaaaa@52DB@ et δ i w c 2 = ( j s i w i j c i j ) 2 ( j s i ( w i j c i j ) 2 / a i j ) . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiTdq2aa0 baaSqaaiaadMgacaWG3bGaam4yaaqaaiaaikdaaaGccqGH9aqpdaqa daqaamaaqababaGaam4DamaaBaaaleaacaWGPbGaamOAaaqabaGcca WGJbWaaSbaaSqaaiaadMgacaWGQbaabeaaaeaacaWGQbGaeyicI4Sa am4CamaaBaaameaacaWGPbaabeaaaSqab0GaeyyeIuoaaOGaayjkai aawMcaamaaCaaaleqabaGaeyOeI0IaaGOmaaaakmaabmaabaWaaSGb aeaadaaeqaqaamaabmaabaGaam4DamaaBaaaleaacaWGPbGaamOAaa qabaGccaWGJbWaaSbaaSqaaiaadMgacaWGQbaabeaaaOGaayjkaiaa wMcaamaaCaaaleqabaGaaGOmaaaaaeaacaWGQbGaeyicI4Saam4Cam aaBaaameaacaWGPbaabeaaaSqab0GaeyyeIuoaaOqaaiaadggadaWg aaWcbaGaamyAaiaadQgaaeqaaaaaaOGaayjkaiaawMcaaiaac6caaa a@607C@ Les termes y ¯ i w c MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmyEayaara WaaSbaaSqaaiaadMgacGaDao4Daiaadogaaeqaaaaa@3B07@ et z ¯ i w c , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabCOEayaara WaaSbaaSqaaiaadMgacaWG3bGaam4yaaqabaGccaGGSaaaaa@3ACA@ sont les moyennes pondérées fondées sur le modèle déjà définies, respectivement pour y MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEaaaa@36F5@ et z . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCOEaiaac6 caaaa@37AC@ Le vecteur de régression β MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCOSdaaa@3735@ est estimé comme ceci :

β ^ PEBLUP = ( i = 1 m j s i w i j a i j ( z i j γ ^ i w a z ¯ i w a ) z i j T ) 1 i = 1 m j s i w i j a i j ( z i j γ ^ i w a z ¯ i w a ) y i j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabCOSdyaaja WaaWbaaSqabeaacaqGqbGaaeyraiaabkeacaqGmbGaaeyvaiaabcfa aaGccqGH9aqpdaqadaqaamaaqahabaWaaabuaeaacaWG3bWaaSbaaS qaaiaadMgacaWGQbaabeaakiaadggadaWgaaWcbaGaamyAaiaadQga aeqaaOWaaeWaaeaacaWH6bWaaSbaaSqaaiaadMgacaWGQbaabeaaki abgkHiTiqbeo7aNzaajaWaaSbaaSqaaiaadMgacaWG3bGaamyyaaqa baGcceWH6bGbaebadaWgaaWcbaGaamyAaiaadEhacaWGHbaabeaaaO GaayjkaiaawMcaaiaahQhadaqhaaWcbaGaamyAaiaadQgaaeaacaWG ubaaaaqaaiaadQgacqGHiiIZcaWGZbWaaSbaaWqaaiaadMgaaeqaaa WcbeqdcqGHris5aaWcbaGaamyAaiabg2da9iaaigdaaeaacaWGTbaa niabggHiLdaakiaawIcacaGLPaaadaahaaWcbeqaaiabgkHiTiaaig daaaGcdaaeWbqaamaaqafabaGaam4DamaaBaaaleaacaWGPbGaamOA aaqabaGccaWGHbWaaSbaaSqaaiaadMgacaWGQbaabeaakmaabmaaba GaaCOEamaaBaaaleaacaWGPbGaamOAaaqabaGccqGHsislcuaHZoWz gaqcamaaBaaaleaacaWGPbGaam4DaiaadggaaeqaaOGabCOEayaara WaaSbaaSqaaiaadMgacaWG3bGaamyyaaqabaaakiaawIcacaGLPaaa caWG5bWaaSbaaSqaaiaadMgacaWGQbaabeaaaeaacaWGQbGaeyicI4 Saam4CamaaBaaameaacaWGPbaabeaaaSqab0GaeyyeIuoaaSqaaiaa dMgacqGH9aqpcaaIXaaabaGaamyBaaqdcqGHris5aaaa@89B7@

z ¯ i w a = ( j s i w i j a i j ) 1 j s i w i j a i j z i j , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabCOEayaara WaaSbaaSqaaiaadMgacaWG3bGaamyyaaqabaGccqGH9aqpdaqadeqa amaaqababaGaam4DamaaBaaaleaacaWGPbGaamOAaaqabaGccaWGHb WaaSbaaSqaaiaadMgacaWGQbaabeaaaeaacaWGQbGaeyicI4Saam4C amaaBaaameaacaWGPbaabeaaaSqab0GaeyyeIuoaaOGaayjkaiaawM caamaaCaaaleqabaGaeyOeI0IaaGymaaaakmaaqababaGaam4Damaa BaaaleaacaWGPbGaamOAaaqabaGccaWGHbWaaSbaaSqaaiaadMgaca WGQbaabeaakiaahQhadaWgaaWcbaGaamyAaiaadQgaaeqaaaqaaiaa dQgacqGHiiIZcaWGZbWaaSbaaWqaaiaadMgaaeqaaaWcbeqdcqGHri s5aOGaaiilaaaa@5B2F@ γ ^ i w a = ( σ ^ v 2 + σ ^ e 2 δ i w a 2 ) 1 σ ^ v 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafq4SdCMbaK aadaWgaaWcbaGaamyAaiaadEhacaWGHbaabeaakiabg2da9maabmqa baGafq4WdmNbaKaadaqhaaWcbaGaamODaaqaaiaaikdaaaGccqGHRa WkcuaHdpWCgaqcamaaDaaaleaacaWGLbaabaGaaGOmaaaakiabes7a KnaaDaaaleaacaWGPbGaam4DaiaadggaaeaacaaIYaaaaaGccaGLOa GaayzkaaWaaWbaaSqabeaacqGHsislcaaIXaaaaOGafq4WdmNbaKaa daqhaaWcbaGaamODaaqaaiaaikdaaaaaaa@5095@ et où δ i w a 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiTdq2aa0 baaSqaaiaadMgacaWG3bGaamyyaaqaaiaaikdaaaaaaa@3B55@ est calculé comme δ i w a 2 = ( j s i w i j a i j ) 2 ( j s i ( w i j a i j ) 2 / a i j ) . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiTdq2aa0 baaSqaaiaadMgacaWG3bGaamyyaaqaaiaaikdaaaGccqGH9aqpdaqa deqaamaaqababaGaam4DamaaBaaaleaacaWGPbGaamOAaaqabaGcca WGHbWaaSbaaSqaaiaadMgacaWGQbaabeaaaeaacaWGQbGaeyicI4Sa am4CamaaBaaameaacaWGPbaabeaaaSqab0GaeyyeIuoaaOGaayjkai aawMcaamaaCaaaleqabaGaeyOeI0IaaGOmaaaakmaabmqabaWaaSGb aeaadaaeqaqaamaabmqabaGaam4DamaaBaaaleaacaWGPbGaamOAaa qabaGccaWGHbWaaSbaaSqaaiaadMgacaWGQbaabeaaaOGaayjkaiaa wMcaamaaCaaaleqabaGaaGOmaaaaaeaacaWGQbGaeyicI4Saam4Cam aaBaaameaacaWGPbaabeaaaSqab0GaeyyeIuoaaOqaaiaadggadaWg aaWcbaGaamyAaiaadQgaaeqaaaaaaOGaayjkaiaawMcaaiaac6caaa a@6079@

Les composantes de la variance, σ e 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aa0 baaSqaaiaadwgaaeaacaaIYaaaaaaa@398D@ et σ v 2 , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aa0 baaSqaaiaadAhaaeaacaaIYaaaaOGaaiilaaaa@3A58@ sont estimées au moyen de l’ajustement des constantes (non pondérées par les poids d’enquête), comme l’ont indiqué Battese et coll. (1988) ou Rao (2003). Les estimateurs de σ e 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aa0 baaSqaaiaadwgaaeaacaaIYaaaaaaa@398D@ qui en découlent sont toujours supérieurs ou égaux à zéro, mais l’estimateur de σ v 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aa0 baaSqaaiaadAhaaeaacaaIYaaaaaaa@399E@ peut être négatif. Si σ v 2 < 0 , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafq4WdmNbam badaqhaaWcbaGaamODaaqaaiaaikdaaaGccqGH8aapcaaIWaGaaiil aaaa@3C30@ il s’établit à zéro, ce qui implique qu’il n’y a pas d’effets sur le domaine. On obtient les EQM connexes en élargissant les méthodes de You et Rao (2002) et de Stukel et Rao (1997).

À noter que, si l’échantillon s MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4Caaaa@36EF@ est sélectionné dans l’univers U , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyvaiaacY caaaa@3781@ la fraction de l’échantillon réalisé, f i = n i / N i , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaaBa aaleaacaWGPbaabeaakiabg2da9maalyaabaGaamOBamaaBaaaleaa caWGPbaabeaaaOqaaiaad6eadaWgaaWcbaGaamyAaaqabaaaaOGaai ilaaaa@3DE0@ pourrait être non négligeable. Pour estimer une moyenne de population, Y ¯ i , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmywayaara WaaSbaaSqaaiaadMgaaeqaaOGaaiilaaaa@38C1@ Rao et Molina (2015) ont pris en compte des fractions d’échantillonnage non négligeables en les exprimant comme suit :

Y ¯ i = f i y ¯ i s + ( 1 f i ) y ¯ is ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmywayaara WaaSbaaSqaaiaadMgaaeqaaOGaeyypa0JaamOzamaaBaaaleaacaWG PbaabeaakiqadMhagaqeamaaBaaaleaacaWGPbGaam4CaaqabaGccq GHRaWkdaqadaqaaiaaigdacqGHsislcaWGMbWaaSbaaSqaaiaadMga aeqaaaGccaGLOaGaayzkaaGabmyEayaaraWaaSbaaSqaaiaadMgace WGZbGbaebaaeqaaaaa@47B9@

y ¯ i s MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmyEayaara WaaSbaaSqaaiaadMgacaWGZbaabeaaaaa@391F@ est la moyenne de l’échantillon du i e MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAamaaCa aaleqabaGaaeyzaaaaaaa@37FA@ domaine échantillonné et y ¯ is ¯ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmyEayaara WaaSbaaSqaaiaadMgaceWGZbGbaebaaeqaaaaa@3937@ est la moyenne de l’échantillon des unités non échantillonnées dans ce domaine. Ils ont établi des prédictions pour y ¯ is ¯ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmyEayaara WaaSbaaSqaaiaadMgaceWGZbGbaebaaeqaaaaa@3937@ à l’aide du modèle au niveau de l’unité obtenu par l’équation (4.1). Leurs expressions correspondent au cas où c i j = 1. MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4yamaaBa aaleaacaWGPbGaamOAaaqabaGccqGH9aqpcaaIXaGaaiOlaaaa@3B65@ Cet estimateur a été élargi par Rubin-Bleuer (2014) afin d’inclure les estimateurs EBLUP et PEBLUP au cas où c i j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4yamaaBa aaleaacaWGPbGaamOAaaqabaaaaa@38E8@ serait arbitraire. Des détails précis qui tiennent également compte de l’estimation de l’EQM se trouvent dans Estevao et coll. (2015).

4.1  Réconciliation

Le système de production actuel ne renferme pas de procédure qui permette de réconcilier les estimations obtenues à l’aide du modèle au niveau de l’unité. Toutefois, on peut modifier la méthode du rajustement de la différence de façon appropriée pour remédier à la situation. Les estimateurs EBLUP et PEBLUP prennent la forme suivante :

θ ^ i EPD = { γ ^ i * y ¯ i * + ( Z ¯ i c γ ^ i * z ¯ i * ) T β ^ EPD si  i A Z ¯ i c T β ^ EPD si  i A ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiUdeNbaK aadaqhaaWcbaGaamyAaaqaaiaabweacaqGqbGaaeiraaaakiabg2da 9maaceaabaqbaeaabiGaaaqaaiqbeo7aNzaajaWaa0baaSqaaiaadM gaaeaacaGGQaaaaOGabmyEayaaraWaa0baaSqaaiaadMgaaeaacaGG QaaaaOGaey4kaSIaaiikaiqahQfagaqeamaaBaaaleaacaWGPbGaam 4yaaqabaGccqGHsislcuaHZoWzgaqcamaaDaaaleaacaWGPbaabaGa aiOkaaaakiqahQhagaqeamaaDaaaleaacaWGPbaabaGaaiOkaaaaki aacMcadaahaaWcbeqaaiaadsfaaaGcceWHYoGbaKaadaahaaWcbeqa aiaabweacaqGqbGaaeiraaaaaOqaaiaabohacaqGPbGaaeiiaiaadM gacqGHiiIZcaWGbbaabaGabCOwayaaraWaa0baaSqaaiaadMgacaWG JbaabaGaamivaaaakiqahk7agaqcamaaCaaaleqabaGaaeyraiaabc facaqGebaaaaGcbaGaae4CaiaabMgacaqGGaGaamyAaiabgIGiolqa dgeagaqeaaaaaiaawUhaaaaa@68DD@

γ ^ i * , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafq4SdCMbaK aadaqhaaWcbaGaamyAaaqaaiaacQcaaaGccaGGSaaaaa@3A31@ y ¯ i * , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmyEayaara Waa0baaSqaaiaadMgaaeaacaGGQaaaaOGaaiilaaaa@3990@ z ¯ i * , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabCOEayaara Waa0baaSqaaiaadMgaaeaacaGGQaaaaOGaaiilaaaa@3995@ et β ^ EPD MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabCOSdyaaja WaaWbaaSqabeaacaqGfbGaaeiuaiaabseaaaaaaa@39D4@ correspondent aux termes définis précédemment : γ ^ i * MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafq4SdCMbaK aadaqhaaWcbaGaamyAaaqaaiaacQcaaaaaaa@3977@ est égal à γ ^ i a MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafq4SdCMbaK aadaWgaaWcbaGaamyAaiaadggaaeqaaaaa@39AE@ pour EBLUP, et à γ ^ i w c MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafq4SdCMbaK aadaWgaaWcbaGaamyAaiaadEhacaWGJbaabeaaaaa@3AAC@ pour PEBLUP; y ¯ i * MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmyEayaara Waa0baaSqaaiaadMgaaeaacaGGQaaaaaaa@38D6@ est égal à y ¯ i a MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmyEayaara WaaSbaaSqaaiaadMgacaWGHbaabeaaaaa@390D@ pour EBLUP, et à y ¯ i w c MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmyEayaara WaaSbaaSqaaiaadMgacGaDao4Daiaadogaaeqaaaaa@3B07@ pour PEBLUP; z ¯ i * MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabCOEayaara Waa0baaSqaaiaadMgaaeaacaGGQaaaaaaa@38DB@ est égal à z ¯ i a MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabCOEayaara WaaSbaaSqaaiaadMgacaWGHbaabeaaaaa@3912@ pour EBLUP, et à z ¯ i w c MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabCOEayaara WaaSbaaSqaaiaadMgacaWG3bGaam4yaaqabaaaaa@3A10@ pour PEBLUP; et, β ^ EPD MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabCOSdyaaja WaaWbaaSqabeaacaqGfbGaaeiuaiaabseaaaaaaa@39D4@ est égal à β ^ EBLUP MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabCOSdyaaja WaaWbaaSqabeaacaqGfbGaaeOqaiaabYeacaqGvbGaaeiuaaaaaaa@3B79@ pour EBLUP, et à β ^ PEBLUP MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabCOSdyaaja WaaWbaaSqabeaacaqGqbGaaeyraiaabkeacaqGmbGaaeyvaiaabcfa aaaaaa@3C4C@ pour PEBLUP.

Supposons qu’il faut réconcilier θ ^ i EPD MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiUdeNbaK aadaqhaaWcbaGaamyAaaqaaiaabweacaqGqbGaaeiraaaaaaa@3B3A@ comme θ ^ * . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiUdeNbaK aadaahaaWcbeqaaiaacQcaaaGccaGGUaaaaa@3954@ L’estimateur réconcilié correspondant est :

θ ^ i EPD , b = { θ ^ i EPD + α i ( θ * d A ω d θ ^ d EPD ) si i A Z ¯ i c T β ^ EPD si i A ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiUdeNbaK aadaqhaaWcbaGaamyAaaqaaiaabweacaqGqbGaaeiraiaacYcacaaM e8UaamOyaaaakiabg2da9maaceaabaqbaeaabiGaaaqaaiqbeI7aXz aajaWaa0baaSqaaiaadMgaaeaacaqGfbGaaeiuaiaabseaaaGccqGH RaWkcqaHXoqydaWgaaWcbaGaamyAaaqabaGcdaqadaqaaiabeI7aXn aaCaaaleqabaGaaiOkaaaakiabgkHiTmaaqafabaGaeqyYdC3aaSba aSqaaiaadsgaaeqaaaqaaiaadsgacqGHiiIZcaWGbbaabeqdcqGHri s5aOGafqiUdeNbaKaadaqhaaWcbaGaamizaaqaaiaabweacaqGqbGa aeiraaaaaOGaayjkaiaawMcaaaqaaiaabohacaqGPbGaaGjbVlaayk W7caWGPbGaeyicI4SaamyqaaqaaiqahQfagaqeamaaDaaaleaacaWG PbGaam4yaaqaaiaadsfaaaGcceWHYoGbaKaadaahaaWcbeqaaiaabw eacaqGqbGaaeiraaaaaOqaaiaabohacaqGPbGaaGjbVlaaykW7caWG PbGaeyicI4SabmyqayaaraaaaaGaay5Eaaaaaa@745C@

α i = ( d A ω d 2 τ d ) 1 ( ω i τ i ) . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySde2aaS baaSqaaiaadMgaaeqaaOGaeyypa0ZaaeWaaeaadaaeqaqaaiabeM8a 3naaDaaaleaacaWGKbaabaGaaGOmaaaakiabes8a0naaBaaaleaaca WGKbaabeaaaeaacaWGKbGaeyicI4Saamyqaaqab0GaeyyeIuoaaOGa ayjkaiaawMcaamaaCaaaleqabaGaeyOeI0IaaGymaaaakmaabmaaba GaeqyYdC3aaSbaaSqaaiaadMgaaeqaaOGaeqiXdq3aaSbaaSqaaiaa dMgaaeqaaaGccaGLOaGaayzkaaGaaiOlaaaa@50D5@ Le terme ω i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyYdC3aaS baaSqaaiaadMgaaeqaaaaa@38DE@ est défini comme suit : ω i = 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyYdC3aaS baaSqaaiaadMgaaeqaaOGaeyypa0JaaGymaaaa@3AA9@ si la réconciliation est pour un total et ω i = N i / N MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyYdC3aaS baaSqaaiaadMgaaeqaaOGaeyypa0ZaaSGbaeaacaWGobWaaSbaaSqa aiaadMgaaeqaaaGcbaGaamOtaaaaaaa@3CCE@ si la réconciliation est pour la moyenne. Les options possibles pour les τ i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiXdq3aaS baaSqaaiaadMgaaeqaaaaa@38D6@ sont σ ^ v 2 + σ ^ e 2 δ i a 2 , δ i a 2 = ( j = 1 n i a i j ) 1 , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafq4WdmNbaK aadaqhaaWcbaGaamODaaqaaiaaikdaaaGccqGHRaWkcuaHdpWCgaqc amaaDaaaleaacaWGLbaabaGaaGOmaaaakiabes7aKnaaDaaaleaaca WGPbGaamyyaaqaaiaaikdaaaGccaGGSaGaaGjbVlabes7aKnaaDaaa leaacaWGPbGaamyyaaqaaiaaikdaaaGccqGH9aqpdaqadaqaamaaqa dabaGaamyyamaaBaaaleaacaWGPbGaamOAaaqabaaabaGaamOAaiab g2da9iaaigdaaeaacaWGUbWaaSbaaWqaaiaadMgaaeqaaaqdcqGHri s5aaGccaGLOaGaayzkaaWaaWbaaSqabeaacqGHsislcaaIXaaaaOGa aiilaaaa@582B@ pour EBLUP, et σ ^ v 2 + σ ^ e 2 δ i w c 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafq4WdmNbaK aadaqhaaWcbaGaamODaaqaaiaaikdaaaGccqGHRaWkcuaHdpWCgaqc amaaDaaaleaacaWGLbaabaGaaGOmaaaakiabes7aKnaaDaaaleaaca WGPbGaam4DaiaadogaaeaacaaIYaaaaaaa@43AA@ pour PEBLUP.

4.2  Estimation de l’erreur quadratique moyenne

Les estimations de l’erreur quadratique moyenne des estimateurs au niveau de l’unité sont fondées sur l’estimation de leur erreur quadratique moyenne, selon le modèle (4.1) et la structure d’erreur e i j ind ( 0 , σ e 2 / a i j ) . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyzamaaBa aaleaacaWGPbGaamOAaaqabaGccaaMe8+aaybyaeqaleqabaGaaeyA aiaab6gacaqGKbaabaqeeuuDJXwAKbsr4rNCHbacfaqcLbwacqWF8i IoaaGccaaMe8+aaeWaaeaacaaIWaGaaiilaiaaysW7daWcgaqaaiab eo8aZnaaDaaaleaacaWGLbaabaGaaGOmaaaaaOqaaiaadggadaWgaa WcbaGaamyAaiaadQgaaeqaaaaaaOGaayjkaiaawMcaaiaac6caaaa@51CF@ Le tableau 4.1 présente ces EQM estimées.


Tableau 4.1
Estimations des EQM pour les estimateurs au niveau de l’unité
Sommaire du tableau
Le tableau montre les résultats de Estimations des EQM pour les estimateurs au niveau de l’unité. Les données sont présentées selon Estimateur (titres de rangée) et EQM(figurant comme en-tête de colonne).
Estimateur EQM
EBLUP eqm( θ ^ i EBLUP )={ g 1ia + g 2ia +2 g 3ia pour  iA Z ¯ i T var( β ^ EBLUP ) Z ¯ i + σ ^ v 2 pour  i A ¯ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeyzaiaabg hacaqGTbWaaeWaaeaacuaH4oqCgaqcamaaDaaaleaacaWGPbaabaGa aeyraiaabkeacaqGmbGaaeyvaiaabcfaaaaakiaawIcacaGLPaaacq GH9aqpdaGabaqaauaabaqaciaaaeaacaWGNbWaaSbaaSqaaiaaigda caWGPbGaamyyaaqabaGccqGHRaWkcaWGNbWaaSbaaSqaaiaaikdaca WGPbGaamyyaaqabaGccqGHRaWkcaaIYaGaam4zamaaBaaaleaacaaI ZaGaamyAaiaadggaaeqaaaGcbaGaaeiCaiaab+gacaqG1bGaaeOCai aabccacaqGGaGaamyAaiabgIGiolaadgeaaeaaceWHAbGbaebadaqh aaWcbaGaamyAaaqaaiaadsfaaaGcciGG2bGaaiyyaiaackhadaqada qaaiqahk7agaqcamaaCaaaleqabaGaaeyraiaabkeacaqGmbGaaeyv aiaabcfaaaaakiaawIcacaGLPaaaceWHAbGbaebadaWgaaWcbaGaam yAaaqabaGccqGHRaWkcuaHdpWCgaqcamaaDaaaleaacaWG2baabaGa aGOmaaaaaOqaaiaabchacaqGVbGaaeyDaiaabkhacaqGGaGaaeiiai aadMgacqGHiiIZceWGbbGbaebaaaaacaGL7baaaaa@7785@
PEBLUP eqm( θ ^ i PEBLUP )={ g 1iw + g 2iw +2 g 3iw pouriA Z ¯ i T var( β ^ PEBLUP ) Z ¯ i + σ ^ v 2 pouri A ¯ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeyzaiaabg hacaqGTbWaaeWaaeaacuaH4oqCgaqcamaaDaaaleaacaWGPbaabaGa aeiuaiaabweacaqGcbGaaeitaiaabwfacaqGqbaaaaGccaGLOaGaay zkaaGaeyypa0ZaaiqaaeaafaqaaeGacaaabaGaam4zamaaBaaaleaa caaIXaGaamyAaiaadEhaaeqaaOGaey4kaSIaam4zamaaBaaaleaaca aIYaGaamyAaiaadEhaaeqaaOGaey4kaSIaaGOmaiaadEgadaWgaaWc baGaaG4maiaadMgacaWG3baabeaaaOqaaiaabchacaqGVbGaaeyDai aabkhacaaMe8UaaGPaVlaadMgacqGHiiIZcaWGbbaabaGabCOwayaa raWaa0baaSqaaiaadMgaaeaacaWGubaaaOGaciODaiaacggacaGGYb WaaeWaaeaaceWHYoGbaKaadaahaaWcbeqaaiaabcfacaqGfbGaaeOq aiaabYeacaqGvbGaaeiuaaaaaOGaayjkaiaawMcaaiqahQfagaqeam aaBaaaleaacaWGPbaabeaakiabgUcaRiqbeo8aZzaajaWaa0baaSqa aiaadAhaaeaacaaIYaaaaaGcbaGaaeiCaiaab+gacaqG1bGaaeOCai aaysW7caaMc8UaamyAaiabgIGiolqadgeagaqeaaaaaiaawUhaaaaa @7D11@

Les divers termes g MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4zaaaa@36E3@ au tableau 4.1 peuvent être interprétés de la même façon que ceux associés aux EQM au niveau du domaine. Les g 1 i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4zamaaBa aaleaacaaIXaGaamyAaaqabaaaaa@38B8@ désignés par g 1 i a MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4zamaaBa aaleaacaaIXaGaamyAaiaadggaaeqaaaaa@399E@ pour EBLUP, et g 1 i w MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4zamaaBa aaleaacaaIXaGaamyAaiaadEhaaeqaaaaa@39B4@ pour PEBLUP représentent la majeure partie des EQM si le nombre de domaines est élevé. Les g 2 i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4zamaaBa aaleaacaaIYaGaamyAaaqabaaaaa@38B9@ représentent l’estimation de β , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCOSdiaacY caaaa@37E5@ et les 2 g 3 i , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOmaiaadE gadaWgaaWcbaGaaG4maiaadMgaaeqaaOGaaiilaaaa@3A30@ l’estimation de σ v 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aa0 baaSqaaiaadAhaaeaacaaIYaaaaaaa@399E@ et σ e 2 . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aa0 baaSqaaiaadwgaaeaacaaIYaaaaOGaaiOlaaaa@3A49@

Les variances estimées de β ^ EBLUP MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabCOSdyaaja WaaWbaaSqabeaacaqGfbGaaeOqaiaabYeacaqGvbGaaeiuaaaaaaa@3B79@ et β ^ PEBLUP MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabCOSdyaaja WaaWbaaSqabeaacaqGqbGaaeyraiaabkeacaqGmbGaaeyvaiaabcfa aaaaaa@3C4C@ sont représentées respectivement par :

var ( β ^ EBLUP ) = σ ^ e 2 ( i A j s i a i j ( z i j γ ^ i a x ¯ i a ) z i j T ) 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbiqaaaGeciGG2b Gaaiyyaiaackhadaqadaqaaiqahk7agaqcamaaCaaaleqabaGaaeyr aiaabkeacaqGmbGaaeyvaiaabcfaaaaakiaawIcacaGLPaaacqGH9a qpcuaHdpWCgaqcamaaDaaaleaacaWGLbaabaGaaGOmaaaakmaabmaa baWaaabuaeaadaaeqbqaaiaadggadaWgaaWcbaGaamyAaiaadQgaae qaaOWaaeWaaeaacaWH6bWaaSbaaSqaaiaadMgacaWGQbaabeaakiab gkHiTiqbeo7aNzaajaWaaSbaaSqaaiaadMgacaWGHbaabeaakiqahI hagaqeamaaBaaaleaacaWGPbGaamyyaaqabaaakiaawIcacaGLPaaa caWH6bWaa0baaSqaaiaadMgacaWGQbaabaGaamivaaaaaeaacaWGQb GaeyicI4Saam4CamaaBaaameaacaWGPbaabeaaaSqab0GaeyyeIuoa aSqaaiaadMgacqGHiiIZcaWGbbaabeqdcqGHris5aaGccaGLOaGaay zkaaWaaWbaaSqabeaacqGHsislcaaIXaaaaaaa@6768@

et

var ( β ^ PEBLUP ) = σ ^ e 2 ( i A j s i z i j * z i j * T ) 1 ( i A j s i z i j * z i j * T / a i j ) ( i A j s i z i j * z i j * T ) 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciODaiaacg gacaGGYbWaaeWaaeaaceWHYoGbaKaadaahaaWcbeqaaiaabcfacaqG fbGaaeOqaiaabYeacaqGvbGaaeiuaaaaaOGaayjkaiaawMcaaiabg2 da9iqbeo8aZzaajaWaa0baaSqaaiaadwgaaeaacaaIYaaaaOWaaeWa aeaadaaeqbqaamaaqafabaGaaCOEamaaDaaaleaacaWGPbGaamOAaa qaaiaacQcaaaGccaWH6bWaa0baaSqaaiaadMgacaWGQbaabaGaaiOk aiaadsfaaaaabaGaamOAaiabgIGiolaadohadaWgaaadbaGaamyAaa qabaaaleqaniabggHiLdaaleaacaWGPbGaeyicI4Saamyqaaqab0Ga eyyeIuoaaOGaayjkaiaawMcaamaaCaaaleqabaGaeyOeI0IaaGymaa aakmaabmaabaWaaSGbaeaadaaeqbqaamaaqafabaGaaCOEamaaDaaa leaacaWGPbGaamOAaaqaaiaacQcaaaGccaWH6bWaa0baaSqaaiaadM gacaWGQbaabaGaaiOkaiaadsfaaaaabaGaamOAaiabgIGiolaadoha daWgaaadbaGaamyAaaqabaaaleqaniabggHiLdaaleaacaWGPbGaey icI4Saamyqaaqab0GaeyyeIuoaaOqaaiaadggadaWgaaWcbaGaamyA aiaadQgaaeqaaaaaaOGaayjkaiaawMcaamaabmaabaWaaabuaeaada aeqbqaaiaahQhadaqhaaWcbaGaamyAaiaadQgaaeaacaGGQaaaaOGa aCOEamaaDaaaleaacaWGPbGaamOAaaqaaiaacQcacaWGubaaaaqaai aadQgacqGHiiIZcaWGZbWaaSbaaWqaaiaadMgaaeqaaaWcbeqdcqGH ris5aaWcbaGaamyAaiabgIGiolaadgeaaeqaniabggHiLdaakiaawI cacaGLPaaadaahaaWcbeqaaiabgkHiTiaaigdaaaaaaa@8DFE@

z i j * = w i j a i j ( z i j γ ^ i w a z ¯ i w a ) . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCOEamaaDa aaleaacaWGPbGaamOAaaqaaiaacQcaaaGccqGH9aqpcaWG3bWaaSba aSqaaiaadMgacaWGQbaabeaakiaadggadaWgaaWcbaGaamyAaiaadQ gaaeqaaOWaaeWaaeaacaWH6bWaaSbaaSqaaiaadMgacaWGQbaabeaa kiabgkHiTiqbeo7aNzaajaWaaSbaaSqaaiaadMgacaWG3bGaamyyaa qabaGcceWH6bGbaebadaWgaaWcbaGaamyAaiac0b4G3bGaamyyaaqa baaakiaawIcacaGLPaaacaGGUaaaaa@50E2@

La forme précise des termes g MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4zaaaa@36E3@ et des variances estimées se trouve dans Estevao et coll. (2015).


Date de modification :