Est-ce que la réduction du déséquilibre de la réponse accroît l’exactitude des estimations de l’enquête ? Section 10. Discussion

Nous commentons ici plusieurs questions qui se posent et indiquons les limites de notre étude.

1. Choix des variables pour le vecteur auxiliaire. Les variables auxiliaires pour le vecteur x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpD0xh9Wqpm0db9Wq pepeuf0xe9q8qiYRWFGCk9vi=dbvc9s8vr0db9Fn0dbbG8Fq0Jfr=x fr=xfbpdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCiEaaaa@3A59@ sont traitées comme un choix fixe dans le présent article. Ce choix est important quand une éventuellement grande quantité de ces variables est disponible. Lesquelles faut-il choisir pour atteindre l’objectif ultime, qui est la production d’estimations les plus exactes possibles ? Le résultat 1 montre que, dans le cas du vecteur de groupes, deux facteurs sont importants pour S Δ 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpD0xh9Wqpm0db9Wq pepeuf0xe9q8qiYRWFGCk9vi=dbvc9s8vr0db9Fn0dbbG8Fq0Jfr=x fr=xfbpdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uamaaDa aaleaacqqHuoaraeaacaaIYaaaaaaa@3C7F@ (qui détermine la variance conditionnelle de CAL): le déséquilibre de la réponse I M B MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpD0xh9Wqpm0db9Wq pepeuf0xe9q8qiYRWFGCk9vi=dbvc9s8vr0db9Fn0dbbG8Fq0Jfr=x fr=xfbpdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysaiaad2 eacaWGcbaaaa@3BBF@ et la variance S y 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpD0xh9Wqpm0db9Wq pepeuf0xe9q8qiYRWFGCk9vi=dbvc9s8vr0db9Fn0dbbG8Fq0Jfr=x fr=xfbpdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uamaaDa aaleaacaWG5baabaGaaGOmaaaaaaa@3C17@ de la variable étudiée y . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpD0xh9Wqpm0db9Wq pepeuf0xe9q8qiYRWFGCk9vi=dbvc9s8vr0db9Fn0dbbG8Fq0Jfr=x fr=xfbpdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEaiaac6 caaaa@3B08@ Le fait que S Δ 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpD0xh9Wqpm0db9Wq pepeuf0xe9q8qiYRWFGCk9vi=dbvc9s8vr0db9Fn0dbbG8Fq0Jfr=x fr=xfbpdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uamaaDa aaleaacqqHuoaraeaacaaIYaaaaaaa@3C7F@ soit (approximativement) linéairement décroissante en I M B MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpD0xh9Wqpm0db9Wq pepeuf0xe9q8qiYRWFGCk9vi=dbvc9s8vr0db9Fn0dbbG8Fq0Jfr=x fr=xfbpdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysaiaad2 eacaWGcbaaaa@3BBF@ incite à essayer de réduire I M B MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpD0xh9Wqpm0db9Wq pepeuf0xe9q8qiYRWFGCk9vi=dbvc9s8vr0db9Fn0dbbG8Fq0Jfr=x fr=xfbpdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysaiaad2 eacaWGcbaaaa@3BBF@ durant la collecte des données. Mais inclure plus de variables dans x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpD0xh9Wqpm0db9Wq pepeuf0xe9q8qiYRWFGCk9vi=dbvc9s8vr0db9Fn0dbbG8Fq0Jfr=x fr=xfbpdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCiEaaaa@3A59@ augmente I M B MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpD0xh9Wqpm0db9Wq pepeuf0xe9q8qiYRWFGCk9vi=dbvc9s8vr0db9Fn0dbbG8Fq0Jfr=x fr=xfbpdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysaiaad2 eacaWGcbaaaa@3BBF@ (parce que l’on recherche une concordance sur un plus grand nombre de moyennes x ) . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpD0xh9Wqpm0db9Wq pepeuf0xe9q8qiYRWFGCk9vi=dbvc9s8vr0db9Fn0dbbG8Fq0Jfr=x fr=xfbpdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeGaaeaaca WH4baacaGLPaaacaGGUaaaaa@3BD3@ Dans le cas de la variance de y , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpD0xh9Wqpm0db9Wq pepeuf0xe9q8qiYRWFGCk9vi=dbvc9s8vr0db9Fn0dbbG8Fq0Jfr=x fr=xfbpdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEaiaacY caaaa@3B06@ S y 2 , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpD0xh9Wqpm0db9Wq pepeuf0xe9q8qiYRWFGCk9vi=dbvc9s8vr0db9Fn0dbbG8Fq0Jfr=x fr=xfbpdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uamaaDa aaleaacaWG5baabaGaaGOmaaaakiaacYcaaaa@3CD1@ c’est le contraire qui se produit. En vertu de (7.1), S y 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpD0xh9Wqpm0db9Wq pepeuf0xe9q8qiYRWFGCk9vi=dbvc9s8vr0db9Fn0dbbG8Fq0Jfr=x fr=xfbpdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uamaaDa aaleaacaWG5baabaGaaGOmaaaaaaa@3C17@ est une variance résiduelle moyenne autour des moyennes de groupe; l’introduction de variables supplémentaires dans x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpD0xh9Wqpm0db9Wq pepeuf0xe9q8qiYRWFGCk9vi=dbvc9s8vr0db9Fn0dbbG8Fq0Jfr=x fr=xfbpdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCiEaaaa@3A59@  réduira S y 2 , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpD0xh9Wqpm0db9Wq pepeuf0xe9q8qiYRWFGCk9vi=dbvc9s8vr0db9Fn0dbbG8Fq0Jfr=x fr=xfbpdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uamaaDa aaleaacaWG5baabaGaaGOmaaaakiaacYcaaaa@3CD1@ surtout si elles expliquent bien y . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpD0xh9Wqpm0db9Wq pepeuf0xe9q8qiYRWFGCk9vi=dbvc9s8vr0db9Fn0dbbG8Fq0Jfr=x fr=xfbpdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEaiaac6 caaaa@3B08@ Les deux facteurs travaillent en sens opposé: un plus grand nombre de variables auxiliaires donne un plus grand déséquilibre I M B , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpD0xh9Wqpm0db9Wq pepeuf0xe9q8qiYRWFGCk9vi=dbvc9s8vr0db9Fn0dbbG8Fq0Jfr=x fr=xfbpdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysaiaad2 eacaWGcbGaaiilaaaa@3C6F@ mais une plus faible variance de y . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpD0xh9Wqpm0db9Wq pepeuf0xe9q8qiYRWFGCk9vi=dbvc9s8vr0db9Fn0dbbG8Fq0Jfr=x fr=xfbpdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEaiaac6 caaaa@3B08@ Cela suggère un compromis possible, question qui n’est pas examinée dans le présent article. L’une des particularités d’un vecteur de groupes x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpD0xh9Wqpm0db9Wq pepeuf0xe9q8qiYRWFGCk9vi=dbvc9s8vr0db9Fn0dbbG8Fq0Jfr=x fr=xfbpdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCiEaaaa@3A59@ joue un rôle, à savoir que, si un plus grand nombre de variables catégoriques entre dans le vecteur, la dimension de celui-ci augmente de façon multiplicative. Le risque que des cellules soient petites ou vides limite l’extension. En guise d’illustration, si x = ( s e x e × é t u d e s × â g e ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpD0xh9Wqpm0db9Wq pepeuf0xe9q8qiYRWFGCk9vi=dbvc9s8vr0db9Fn0dbbG8Fq0Jfr=x fr=xfbpdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCiEaiabg2 da9maabmaabaGaam4CaiaadwgacaWG4bGaamyzaiabgEna0kaadMoa caWG0bGaamyDaiaadsgacaWGLbGaam4CaiabgEna0kaadkoacaWGNb GaamyzaaGaayjkaiaawMcaaaaa@4E48@ de dimension J = 2 × 3 × 4 = 24 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpD0xh9Wqpm0db9Wq pepeuf0xe9q8qiYRWFGCk9vi=dbvc9s8vr0db9Fn0dbbG8Fq0Jfr=x fr=xfbpdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOsaiabg2 da9iaaikdacqGHxdaTcaaIZaGaey41aqRaaGinaiabg2da9iaaikda caaI0aaaaa@4412@ est étendu afin d’inclure également la profession avec 4 catégories, de manière entièrement croisée, la nouvelle dimension (égale au nouveau nombre de groupes) est J = 24 × 4 = 96. MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpD0xh9Wqpm0db9Wq pepeuf0xe9q8qiYRWFGCk9vi=dbvc9s8vr0db9Fn0dbbG8Fq0Jfr=x fr=xfbpdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOsaiabg2 da9iaaikdacaaI0aGaey41aqRaaGinaiabg2da9iaaiMdacaaI2aGa aiOlaaaa@42B7@ En principe, S y 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpD0xh9Wqpm0db9Wq pepeuf0xe9q8qiYRWFGCk9vi=dbvc9s8vr0db9Fn0dbbG8Fq0Jfr=x fr=xfbpdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uamaaDa aaleaacaWG5baabaGaaGOmaaaaaaa@3C17@ diminue, mais le risque d’obtenir de petites cellules est une bonne raison de s’abstenir de croiser entièrement toutes les variables et de les faire plutôt intervenir dans un vecteur x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpD0xh9Wqpm0db9Wq pepeuf0xe9q8qiYRWFGCk9vi=dbvc9s8vr0db9Fn0dbbG8Fq0Jfr=x fr=xfbpdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCiEaaaa@3A59@ sans groupes. Ce cas est abordé dans le résultat 2, selon lequel, si x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpD0xh9Wqpm0db9Wq pepeuf0xe9q8qiYRWFGCk9vi=dbvc9s8vr0db9Fn0dbbG8Fq0Jfr=x fr=xfbpdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCiEaaaa@3A59@ explique bien y , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpD0xh9Wqpm0db9Wq pepeuf0xe9q8qiYRWFGCk9vi=dbvc9s8vr0db9Fn0dbbG8Fq0Jfr=x fr=xfbpdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEaiaacY caaaa@3B06@ alors σ ε 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpD0xh9Wqpm0db9Wq pepeuf0xe9q8qiYRWFGCk9vi=dbvc9s8vr0db9Fn0dbbG8Fq0Jfr=x fr=xfbpdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aa0 baaSqaaiabew7aLbqaaiaaikdaaaaaaa@3DAB@ est petite et donnera une faible variance souhaitée pour Δ r . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpD0xh9Wqpm0db9Wq pepeuf0xe9q8qiYRWFGCk9vi=dbvc9s8vr0db9Fn0dbbG8Fq0Jfr=x fr=xfbpdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuiLdq0aaS baaSqaaiaadkhaaeqaaOGaaiOlaaaa@3C9D@

2. Information auxiliaire à différents niveaux. Dans le présent article, le déséquilibre I M B MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpD0xh9Wqpm0db9Wq pepeuf0xe9q8qiYRWFGCk9vi=dbvc9s8vr0db9Fn0dbbG8Fq0Jfr=x fr=xfbpdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysaiaad2 eacaWGcbaaaa@3BBF@ et l’estimateur par calage Y ^ C A L MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpD0xh9Wqpm0db9Wq pepeuf0xe9q8qiYRWFGCk9vi=dbvc9s8vr0db9Fn0dbbG8Fq0Jfr=x fr=xfbpdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmywayaaja WaaSbaaSqaaiaadoeacaWGbbGaamitaaqabaaaaa@3CD1@ utilisent le même vecteur x , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpD0xh9Wqpm0db9Wq pepeuf0xe9q8qiYRWFGCk9vi=dbvc9s8vr0db9Fn0dbbG8Fq0Jfr=x fr=xfbpdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCiEaiaacY caaaa@3B09@ et plus particulièrement un vecteur qui contient des données auxiliaires pour les unités de l’échantillon uniquement. Il s’agit d’un cas réaliste. Cependant, dans des formulations plus générales, la collecte des données ferait usage d’un vecteur de surveillance x M V MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpD0xh9Wqpm0db9Wq pepeuf0xe9q8qiYRWFGCk9vi=dbvc9s8vr0db9Fn0dbbG8Fq0Jfr=x fr=xfbpdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCiEamaaBa aaleaacaWGnbGaamOvaaqabaaaaa@3C32@ éventuellement différent du vecteur de calage x C A L MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpD0xh9Wqpm0db9Wq pepeuf0xe9q8qiYRWFGCk9vi=dbvc9s8vr0db9Fn0dbbG8Fq0Jfr=x fr=xfbpdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCiEamaaBa aaleaacaWGdbGaamyqaiaadYeaaeqaaaaa@3CE4@ utilisé plus tard dans l’estimation. Le premier est un instrument destiné à obtenir un faible déséquilibre I M B MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpD0xh9Wqpm0db9Wq pepeuf0xe9q8qiYRWFGCk9vi=dbvc9s8vr0db9Fn0dbbG8Fq0Jfr=x fr=xfbpdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysaiaad2 eacaWGcbaaaa@3BBF@ dans la réponse, tandis que le second sert à obtenir de bons poids calés pour Y ^ C A L . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpD0xh9Wqpm0db9Wq pepeuf0xe9q8qiYRWFGCk9vi=dbvc9s8vr0db9Fn0dbbG8Fq0Jfr=x fr=xfbpdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmywayaaja WaaSbaaSqaaiaadoeacaWGbbGaamitaaqabaGccaGGUaaaaa@3D8D@ Une raison pour laquelle x M V MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpD0xh9Wqpm0db9Wq pepeuf0xe9q8qiYRWFGCk9vi=dbvc9s8vr0db9Fn0dbbG8Fq0Jfr=x fr=xfbpdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCiEamaaBa aaleaacaWGnbGaamOvaaqabaaaaa@3C32@ et x C A L MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpD0xh9Wqpm0db9Wq pepeuf0xe9q8qiYRWFGCk9vi=dbvc9s8vr0db9Fn0dbbG8Fq0Jfr=x fr=xfbpdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCiEamaaBa aaleaacaWGdbGaamyqaiaadYeaaeqaaaaa@3CE4@ pourraient différer en pratique est que les variables auxiliaires pour l’estimation peuvent être des versions mises à jour des mêmes variables disponibles au moment de la collecte des données. Il peut y avoir d’autres raisons de choisir x M V MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpD0xh9Wqpm0db9Wq pepeuf0xe9q8qiYRWFGCk9vi=dbvc9s8vr0db9Fn0dbbG8Fq0Jfr=x fr=xfbpdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCiEamaaBa aaleaacaWGnbGaamOvaaqabaaaaa@3C32@ et x C A L MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpD0xh9Wqpm0db9Wq pepeuf0xe9q8qiYRWFGCk9vi=dbvc9s8vr0db9Fn0dbbG8Fq0Jfr=x fr=xfbpdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCiEamaaBa aaleaacaWGdbGaamyqaiaadYeaaeqaaaaa@3CE4@ de sorte qu’ils soient différents. En outre, ils peuvent contenir de l’information (si elle est disponible) au niveau de la population. Des extensions de notre approche à ce genre de situations sont possibles.

3. Avantage incertain d’une réduction du déséquilibre. Schouten et coll. (2014) dégagent des preuves que l’équilibrage de la réponse réduit le biais. Nous constatons aussi qu’il existe une motivation à s’efforcer d’obtenir, durant la collecte des données, un ensemble ultime de réponses dont le déséquilibre I M B MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpD0xh9Wqpm0db9Wq pepeuf0xe9q8qiYRWFGCk9vi=dbvc9s8vr0db9Fn0dbbG8Fq0Jfr=x fr=xfbpdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysaiaad2 eacaWGcbaaaa@3BBF@ est faible. Comme le montrent théoriquement les résultats 1 et 2, et comme le confirment empiriquement les situations de test 1 et 2, un faible déséquilibre donne un écart Y ^ C A L Y ^ F U L = N ^ Δ r MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpD0xh9Wqpm0db9Wq pepeuf0xe9q8qiYRWFGCk9vi=dbvc9s8vr0db9Fn0dbbG8Fq0Jfr=x fr=xfbpdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmywayaaja WaaSbaaSqaaiaadoeacaWGbbGaamitaaqabaGccqGHsislceWGzbGb aKaadaWgaaWcbaGaamOraiaadwfacaWGmbaabeaakiabg2da9iqad6 eagaqcaiabfs5aenaaBaaaleaacaWGYbaabeaaaaa@45D4@ dont l’espérance est nulle ou quasi nulle et dont la variance est petite. Il s’agit de notre protection contre un biais important. Si I M B MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpD0xh9Wqpm0db9Wq pepeuf0xe9q8qiYRWFGCk9vi=dbvc9s8vr0db9Fn0dbbG8Fq0Jfr=x fr=xfbpdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysaiaad2 eacaWGcbaaaa@3BBF@ augmentait, la variance aurait tendance à augmenter. L’espérance nulle de l’écart Y ^ C A L Y ^ F U L MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpD0xh9Wqpm0db9Wq pepeuf0xe9q8qiYRWFGCk9vi=dbvc9s8vr0db9Fn0dbbG8Fq0Jfr=x fr=xfbpdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmywayaaja WaaSbaaSqaaiaadoeacaWGbbGaamitaaqabaGccqGHsislceWGzbGb aKaadaWgaaWcbaGaamOraiaadwfacaWGmbaabeaaaaa@4158@ est une propriété moyenne. Il n’y a aucune garantie que l’écart soit petit pour tout ensemble de répondants particulier r MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpD0xh9Wqpm0db9Wq pepeuf0xe9q8qiYRWFGCk9vi=dbvc9s8vr0db9Fn0dbbG8Fq0Jfr=x fr=xfbpdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCaaaa@3A4F@ présentant un faible I M B . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpD0xh9Wqpm0db9Wq pepeuf0xe9q8qiYRWFGCk9vi=dbvc9s8vr0db9Fn0dbbG8Fq0Jfr=x fr=xfbpdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysaiaad2 eacaWGcbGaaiOlaaaa@3C71@

4. L’équilibre parfait n’élimine pas le biais. Un déséquilibre nul, I M B = 0 , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpD0xh9Wqpm0db9Wq pepeuf0xe9q8qiYRWFGCk9vi=dbvc9s8vr0db9Fn0dbbG8Fq0Jfr=x fr=xfbpdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysaiaad2 eacaWGcbGaeyypa0JaaGimaiaacYcaaaa@3E2F@ implique une égalité des moyennes pour l’ensemble de répondants et l’échantillon complet, x ¯ r = x ¯ s . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpD0xh9Wqpm0db9Wq pepeuf0xe9q8qiYRWFGCk9vi=dbvc9s8vr0db9Fn0dbbG8Fq0Jfr=x fr=xfbpdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabCiEayaara WaaSbaaSqaaiaadkhaaeqaaOGaeyypa0JabCiEayaaraWaaSbaaSqa aiaadohaaeqaaOGaaiOlaaaa@3F9D@ Si cet équilibre parfait était atteint, le terme d’ajustement du biais dans (5.2) serait nul; l’estimateur par calage (CAL) et l’estimateur par facteur d’extension (EXP) sont identiquement égaux. On peut dire que, si un équilibre parfait est atteint, la puissance du vecteur auxiliaire est épuisée, non pas en ce qui concerne sa possibilité d’expliquer la variable étudiée, mais en ce qui concerne sa possibilité de se distinguer de l’estimateur EXP rudimentaire, qui, même s’il n’utilise aucune information auxiliaire, est aussi bon que l’estimateur CAL qui, autrement, est meilleur. Cependant, CAL EXP MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpD0xh9Wqpm0db9Wq pepeuf0xe9q8qiYRWFGCk9vi=dbvc9s8vr0db9Fn0dbbG8Fq0Jfr=x fr=xfbpdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaae4qaiaabg eacaqGmbGaeyyyIORaaeyraiaabIfacaqGqbaaaa@3FF0@ n’est toujours pas une situation idéale. Comme le montre le résultat 1, la variance de l’écart de CAL n’est pas proche de zéro même si le déséquilibre I M B MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpD0xh9Wqpm0db9Wq pepeuf0xe9q8qiYRWFGCk9vi=dbvc9s8vr0db9Fn0dbbG8Fq0Jfr=x fr=xfbpdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysaiaad2 eacaWGcbaaaa@3BBF@  est quasi nul. L’équilibre parfait n’élimine pas l’écart de CAL, mais un petit I M B MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpD0xh9Wqpm0db9Wq pepeuf0xe9q8qiYRWFGCk9vi=dbvc9s8vr0db9Fn0dbbG8Fq0Jfr=x fr=xfbpdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysaiaad2 eacaWGcbaaaa@3BBF@  protège contre un grand écart.

5. Implications pratiques. Dans le présent article, nous nous intéressons principalement aux enquêtes présentant une « non-réponse importante et inévitable » qui ne peut pas raisonnablement (compte tenu des contraintes de temps et de budget de l’enquête) être réduite à un pourcentage à un chiffre, même en engageant d’importantes ressources. Les enquêtes où la non-réponse est égale ou supérieure à 30 % sont fréquentes aujourd’hui. On est loin de la situation idéale où la réponse est quasi totale, et où le déséquilibre et la non-réponse cesseraient essentiellement de poser problème, vu que les estimateurs EXP, CAL et FUL seraient proches les uns des autres.

6. Indications pour la généralisation. Les résultats 1 et 2 montrent les propriétés de l’écart de l’estimateur CAL parmi les ensembles de réponses sous une formulation donnée du vecteur auxiliaire. Il serait souhaitable de généraliser les résultats à d’autres situations. Nos preuves reposent sur l’hypothèse qu’il existe certaines matrices inverses. Les extensions à d’autres cas seraient possibles à l’aide de l’inverse généralisée de Moore-Penrose.

Remerciements

Les présents travaux ont été financés par la subvention 9127 de l’Estonian Science Foundation et par l’Institutional Research Funding IUT34-5 de l’Estonie. Les auteurs tiennent à remercier un rédacteur associé et un examinateur, tous deux anonymes, de leurs commentaires constructifs.

Annexe 1

Obtention du résultat 1

Nous obtenons les expressions (7.2) à (7.4) sous les conditions et la notation exposées à la section 7. L’échantillon s MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fpC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4Caaaa@3516@ est autopondéré, de taille n , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fpC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaiaacY caaaa@35C1@ et x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fpC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCiEaaaa@351F@ est un vecteur de groupes de dimension J . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fpC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOsaiaac6 caaaa@359F@ Nous supposons la probabilité ( n m ) 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFD0xh9Wqpm0db9Wq pepeuf0xe9q8qiYRWFGCk9vi=dbvc9s8vr0db9Fn0dbbG8Fq0Jfr=x fr=xfbpdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaafa qabeGabaaabaqcLbqacaWGUbaakeaajugabiaad2gaaaaakiaawIca caGLPaaadaahaaWcbeqaaiabgkHiTiaaigdaaaaaaa@3F9B@ pour chaque ensemble de répondants r MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fpC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCaaaa@3515@ de taille fixe m . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fpC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBaiaac6 caaaa@35C2@ Nous calculons l’espérance et la variance de Δ r = ( b r b s ) x ¯ s = j = 1 J W j s y ¯ r j y ¯ s , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fpC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuiLdq0aaS baaSqaaiaadkhaaeqaaOGaeyypa0ZaaeWaaeaacaWHIbWaaSbaaSqa aiaadkhaaeqaaOGaeyOeI0IaaCOyamaaBaaaleaacaWGZbaabeaaaO GaayjkaiaawMcaamaaCaaaleqabaGccWaGyBOmGikaaiqahIhagaqe amaaBaaaleaacaWGZbaabeaakiabg2da9maaqadabaGaam4vamaaBa aaleaacaWGQbGaam4CaaqabaaabaGaamOAaiabg2da9iaaigdaaeaa caWGkbaaniabggHiLdGcceWG5bGbaebadaWgaaWcbaGaamOCamaaBa aameaacaWGQbaabeaaaSqabaGccqGHsislceWG5bGbaebadaWgaaWc baGaam4CaaqabaGccaGGSaaaaa@547C@ W j s = n j / n , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fpC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4vamaaBa aaleaacaWGQbGaam4CaaqabaGccqGH9aqpdaWcgaqaaiaad6gadaWg aaWcbaGaamOAaaqabaaakeaacaWGUbaaaiaacYcaaaa@3BEE@ conditionnellement à m MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fpC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBaaaa@3510@ fixe et à la moyenne x ¯ r = ( 1 / m ) ( m 1 , , m j , , m J ) ; MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fpC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabCiEayaara WaaSbaaSqaaiaadkhaaeqaaKaaGjabg2da9OWaaeWaaeaadaWcgaqa aiaaigdaaeaacaWGTbaaaaGaayjkaiaawMcaamaabmaabaqcaaMaam yBaOWaaSbaaKqaGfaacaaIXaaabeaajaaycaGGSaGccqWIMaYsjaay caGGSaGaamyBaOWaaSbaaKqaGfaacaWGQbaabeaajaaycaGGSaGccq WIMaYsjaaycaGGSaGaamyBaOWaaSbaaKqaGfaacaWGkbaabeaaaOGa ayjkaiaawMcaaKaaGjaacUdaaaa@4C0D@ j = 1 J m j = m . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fpC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaabmaeaaca WGTbWaaSbaaSqaaiaadQgaaeqaaaqaaiaadQgacqGH9aqpcaaIXaaa baGaamOsaaqdcqGHris5aOGaeyypa0JaamyBaiaac6caaaa@3E55@ Sous ce conditionnement, R = j = 1 J ( n j m j ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpD0xh9Wqpm0db9Wq pepeuf0xe9q8qiYRWFGCk9vi=dbvc9s8vr0db9Fn0dbbG8Fq0Jfr=x fr=xfbpdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOuaiabg2 da9maaradabaWaaeWaaeaafaqabeGabaaabaqcLbqacaWGUbGcdaWg aaWcbaqcLbkacaWGQbaaleqaaaGcbaqcLbqacaWGTbGcdaWgaaWcba qcLbkacaWGQbaaleqaaaaaaOGaayjkaiaawMcaaaWcbaGaamOAaiab g2da9iaaigdaaeaacaWGkbaaniabg+Givdaaaa@498E@ ensembles r MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fpC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCaaaa@3515@ ont la même probabilité, où n j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fpC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcaaMaamOBaO WaaSbaaKqaGfaacaWGQbaabeaaaaa@36FE@ est la taille du groupe d’échantillon s j ; MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fpC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcaaMaam4CaO WaaSbaaKqaGfaacaWGQbaabeaakiaacUdaaaa@37CC@ j = 1 J n j = n . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fpC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaabmaeaaca WGUbWaaSbaaSqaaiaadQgaaeqaaOGaeyypa0JaamOBaaWcbaGaamOA aiabg2da9iaaigdaaeaacaWGkbaaniabggHiLdGccaGGUaaaaa@3E6C@ Cela est identique à la structure de probabilité pour l’échantillonnage aléatoire simple stratifié de m j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fpC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBamaaBa aaleaacaWGQbaabeaaaaa@362B@ à partir de n j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fpC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBamaaBa aaleaacaWGQbaabeaaaaa@362C@ dans la strate j ; j = 1 , , J . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fpC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOAaiaacU dacaWGQbGaeyypa0JaaGymaiaacYcacqWIMaYscaGGSaGaamOsaiaa c6caaaa@3C7F@ Sachant m MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fpC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBaaaa@3510@ et x ¯ r , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fpC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabCiEayaara WaaSbaaSqaaiaadkhaaeqaaOGaaiilaaaa@3714@ l’espérance et la variance de y ¯ r j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fpC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmyEayaara WaaSbaaSqaaiaadkhadaWgaaadbaGaamOAaaqabaaaleqaaaaa@377E@ sont, respectivement, y ¯ s j = s j y k / n j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fpC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcaaQabmyEay aaraGcdaWgaaqcbaAaaiaadohalmaaBaaajiaObaGaamOAaaqabaaa jeaObeaajaaOcqGH9aqpkmaalyaabaWaaabeaKaaGgaacaWG5bGcda WgaaqcbaAaaiaadUgaaeqaaaqaaiaadohalmaaBaaajiaObaGaamOA aaqabaaajeaObeqcdaQaeyyeIuoaaOqaaiaad6gadaWgaaWcbaGaam OAaaqabaaaaaaa@474C@ et ( 1 / m j 1 / n j ) S y j 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fpC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaada WcgaqaaiaaigdaaeaacaWGTbWaaSbaaSqaaiaadQgaaeqaaaaakiab gkHiTmaalyaabaGaaGymaaqaaiaad6gadaWgaaWcbaGaamOAaaqaba aaaaGccaGLOaGaayzkaaGaam4uamaaDaaaleaacaWG5bGaamOAaaqa aiaaikdaaaaaaa@4013@ avec S y j 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fpC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcaaMaam4uaO Waa0baaKqaGfaacaWG5bGaaGzaVlaadQgaaeaacaaIYaaaaaaa@3A28@ donné en (7.1). Donc, Δ ¯ = j = 1 J W j s y ¯ s j y ¯ s = 0 , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fpC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafuiLdqKbae bacqGH9aqpdaaeWaqaaiaadEfadaWgaaWcbaGaamOAaiaadohaaeqa aaqaaiaadQgacqGH9aqpcaaIXaaabaGaamOsaaqdcqGHris5aOGabm yEayaaraWaaSbaaSqaaiaadohadaWgaaadbaGaamOAaaqabaaaleqa aOGaeyOeI0IabmyEayaaraWaaSbaaSqaaiaadohaaeqaaOGaeyypa0 JaaGimaiaacYcaaaa@481D@ ce qui prouve (7.2), et S Δ 2 = j = 1 J W j s 2 ( 1 / m j 1 / n j ) S y j 2 . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fpC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uamaaDa aaleaacqqHuoaraeaacaaIYaaaaOGaeyypa0ZaaabmaeaacaWGxbWa a0baaSqaaiaadQgacaWGZbaabaGaaGOmaaaaaeaacaWGQbGaeyypa0 JaaGymaaqaaiaadQeaa0GaeyyeIuoakmaabmaabaWaaSGbaeaacaaI XaaabaGaamyBamaaBaaaleaacaWGQbaabeaaaaGccqGHsisldaWcga qaaiaaigdaaeaacaWGUbWaaSbaaSqaaiaadQgaaeqaaaaaaOGaayjk aiaawMcaaiaadofadaqhaaWcbaGaamyEaiaadQgaaeaacaaIYaaaaO GaaiOlaaaa@4E32@ En substituant p j = m j / n j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fpC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCamaaBa aaleaacaWGQbaabeaakiabg2da9maalyaabaGaamyBamaaBaaaleaa caWGQbaabeaaaOqaaiaad6gadaWgaaWcbaGaamOAaaqabaaaaaaa@3B79@ et p = m / n , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fpC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCaiabg2 da9maalyaabaGaamyBaaqaaiaad6gaaaGaaiilaaaa@38C4@ et en utilisant S y 2 = j = 1 J W j s S y j 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fpC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uamaaDa aaleaacaWG5baabaGaaGOmaaaakiabg2da9maaqadabaGaam4vamaa BaaaleaacaWGQbGaam4CaaqabaGccaWGtbWaa0baaSqaaiaadMhaca WGQbaabaGaaGOmaaaaaeaacaWGQbGaeyypa0JaaGymaaqaaiaadQea a0GaeyyeIuoaaaa@440A@ donné en (7.1), nous obtenons

S Δ 2 = 1 n j = 1 J W j s ( 1 p j 1 ) S y j 2 = ( 1 m 1 n ) S y 2 + 1 m j = 1 J W j s ( p p j 1 ) S y j 2 . ( A .1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fpC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uamaaDa aaleaacqqHuoaraeaacaaIYaaaaOGaeyypa0ZaaSaaaeaacaaIXaaa baGaamOBaaaadaaeWbqaaiaadEfadaWgaaWcbaGaamOAaiaadohaae qaaOWaaeWaaeaadaWcaaqaaiaaigdaaeaacaWGWbWaaSbaaSqaaiaa dQgaaeqaaaaakiabgkHiTiaaigdaaiaawIcacaGLPaaacaWGtbWaa0 baaSqaaiaadMhacaWGQbaabaGaaGOmaaaaaeaacaWGQbGaeyypa0Ja aGymaaqaaiaadQeaa0GaeyyeIuoakiabg2da9maabmaabaWaaSaaae aacaaIXaaabaGaamyBaaaacqGHsisldaWcaaqaaiaaigdaaeaacaWG UbaaaaGaayjkaiaawMcaaiaadofadaqhaaWcbaGaamyEaaqaaiaaik daaaGccqGHRaWkdaWcaaqaaiaaigdaaeaacaWGTbaaamaaqahabaGa am4vamaaBaaaleaacaWGQbGaam4CaaqabaGcdaqadaqaamaalaaaba GaamiCaaqaaiaadchadaWgaaWcbaGaamOAaaqabaaaaOGaeyOeI0Ia aGymaaGaayjkaiaawMcaaiaadofadaqhaaWcbaGaamyEaiaadQgaae aacaaIYaaaaOGaaiOlaaWcbaGaamOAaiabg2da9iaaigdaaeaacaWG kbaaniabggHiLdGccaaMf8UaaGzbVlaaywW7caaMf8UaaGzbVlaacI cacaqGbbGaaiOlaiaaigdacaGGPaaaaa@77BC@

Cela prouve (7.3). Pour analyser le terme de pénalisation (deuxième terme du deuxième membre) dans (A.1), supposons que les p j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fpC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcaaMaamiCaO WaaSbaaKqaGfaacaWGQbaabeaaaaa@3700@ ne varient que peu autour du taux global p . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fpC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCaiaac6 caaaa@35C5@ Alors, δ j = p j / p 1 , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fpC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiTdq2aaS baaSqaaiaadQgaaeqaaOGaeyypa0ZaaSGbaeaacaWGWbWaaSbaaSqa aiaadQgaaeqaaaGcbaGaamiCaaaacqGHsislcaaIXaGaaiilaaaa@3D6B@ j = 1 , , J , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fpC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOAaiabg2 da9iaaigdacaGGSaGaeSOjGSKaaiilaiaadQeacaGGSaaaaa@3ACF@ sont de petites quantités, et 1 / p j = 1 / p ( 1 + δ j ) = ( 1 / p ) ( 1 δ j + δ j 2 δ j 3 + ) . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fpC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSGbaeaaca aIXaaabaGaamiCamaaBaaaleaacaWGQbaabeaaaaGccqGH9aqpdaWc gaqaaiaaigdaaeaacaWGWbWaaeWaaeaacaaIXaGaey4kaSIaeqiTdq 2aaSbaaSqaaiaadQgaaeqaaaGccaGLOaGaayzkaaaaaiabg2da9maa bmaabaWaaSGbaeaacaaIXaaabaGaamiCaaaaaiaawIcacaGLPaaada qadaqaaiaaigdacqGHsislcqaH0oazdaWgaaWcbaGaamOAaaqabaGc cqGHRaWkcqaH0oazdaqhaaWcbaGaamOAaaqaaiaaikdaaaGccqGHsi slcqaH0oazdaqhaaWcbaGaamOAaaqaaiaaiodaaaGccqGHRaWkcqWI MaYsaiaawIcacaGLPaaacaGGUaaaaa@55A9@ En gardant les termes jusqu’à l’ordre deux, p / p j 1 δ j + δ j 2 . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fpC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSGbaeaaca WGWbaabaGaamiCamaaBaaaleaacaWGQbaabeaaaaGccqGHsislcaaI XaGaeyisISRaeyOeI0IaeqiTdq2aaSbaaSqaaiaadQgaaeqaaOGaey 4kaSIaeqiTdq2aa0baaSqaaiaadQgaaeaacaaIYaaaaOGaaiOlaaaa @436E@ Le terme de pénalisation est alors approximé par

1 m j = 1 J W j s ( p p j 1 ) S y j 2 1 m j = 1 J W j s ( p j p 1 ) S y j 2 + 1 m j = 1 J W j s ( p j p 1 ) 2 S y j 2 . ( A .2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fpC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aIXaaabaGaamyBaaaadaaeWbqaaiaadEfadaWgaaWcbaGaamOAaiaa dohaaeqaaOWaaeWaaeaadaWcaaqaaiaadchaaeaacaWGWbWaaSbaaS qaaiaadQgaaeqaaaaakiabgkHiTiaaigdaaiaawIcacaGLPaaacaWG tbWaa0baaSqaaiaadMhacaWGQbaabaGaaGOmaaaaaeaacaWGQbGaey ypa0JaaGymaaqaaiaadQeaa0GaeyyeIuoakiabgIKi7kabgkHiTmaa laaabaGaaGymaaqaaiaad2gaaaWaaabCaeaacaWGxbWaaSbaaSqaai aadQgacaWGZbaabeaakmaabmaabaWaaSaaaeaacaWGWbWaaSbaaSqa aiaadQgaaeqaaaGcbaGaamiCaaaacqGHsislcaaIXaaacaGLOaGaay zkaaGaam4uamaaDaaaleaacaWG5bGaamOAaaqaaiaaikdaaaaabaGa amOAaiabg2da9iaaigdaaeaacaWGkbaaniabggHiLdGccqGHRaWkda WcaaqaaiaaigdaaeaacaWGTbaaamaaqahabaGaam4vamaaBaaaleaa caWGQbGaam4CaaqabaGcdaqadaqaamaalaaabaGaamiCamaaBaaale aacaWGQbaabeaaaOqaaiaadchaaaGaeyOeI0IaaGymaaGaayjkaiaa wMcaamaaCaaaleqabaGaaGOmaaaakiaadofadaqhaaWcbaGaamyEai aadQgaaeaacaaIYaaaaaqaaiaadQgacqGH9aqpcaaIXaaabaGaamOs aaqdcqGHris5aOGaaiOlaiaaywW7caaMf8UaaGzbVlaaywW7caaMf8 UaaiikaiaabgeacaqGUaGaaeOmaiaacMcaaaa@81E7@

Supposons en outre que les variances de groupe S y j 2 , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fpC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uamaaDa aaleaacaWG5bGaamOAaaqaaiaaikdaaaGccaGGSaaaaa@3886@ j = 1 , , J MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fpC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOAaiabg2 da9iaaigdacaGGSaGaeSOjGSKaaiilaiaadQeaaaa@3A1F@ ne varient que peu autour de leur moyenne pondérée S y 2 . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fpC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uamaaDa aaleaacaWG5baabaGaaGOmaaaakiaac6caaaa@3799@ En utilisant l’approximation S y j 2 S y 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fpC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uamaaDa aaleaacaWG5bGaamOAaaqaaiaaikdaaaGccqGHijYUcaWGtbWaa0ba aSqaaiaadMhaaeaacaaIYaaaaaaa@3C46@ dans (A.2), nous obtenons

S y 2 m j = 1 J W j s ( p p j 1 ) S y 2 m j = 1 J W j s ( p j p 1 ) + S y 2 m j = 1 J W j s ( p j p 1 ) 2 . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fpC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca WGtbWaa0baaSqaaiaadMhaaeaacaaIYaaaaaGcbaGaamyBaaaadaae WbqaaiaadEfadaWgaaWcbaGaamOAaiaadohaaeqaaOWaaeWaaeaada WcaaqaaiaadchaaeaacaWGWbWaaSbaaSqaaiaadQgaaeqaaaaakiab gkHiTiaaigdaaiaawIcacaGLPaaaaSqaaiaadQgacqGH9aqpcaaIXa aabaGaamOsaaqdcqGHris5aOGaeyisISRaeyOeI0YaaSaaaeaacaWG tbWaa0baaSqaaiaadMhaaeaacaaIYaaaaaGcbaGaamyBaaaadaaeWb qaaiaadEfadaWgaaWcbaGaamOAaiaadohaaeqaaOWaaeWaaeaadaWc aaqaaiaadchadaWgaaWcbaGaamOAaaqabaaakeaacaWGWbaaaiabgk HiTiaaigdaaiaawIcacaGLPaaaaSqaaiaadQgacqGH9aqpcaaIXaaa baGaamOsaaqdcqGHris5aOGaey4kaSYaaSaaaeaacaWGtbWaa0baaS qaaiaadMhaaeaacaaIYaaaaaGcbaGaamyBaaaadaaeWbqaaiaadEfa daWgaaWcbaGaamOAaiaadohaaeqaaOWaaeWaaeaadaWcaaqaaiaadc hadaWgaaWcbaGaamOAaaqabaaakeaacaWGWbaaaiabgkHiTiaaigda aiaawIcacaGLPaaadaahaaWcbeqaaiaaikdaaaaabaGaamOAaiabg2 da9iaaigdaaeaacaWGkbaaniabggHiLdGccaGGUaaaaa@71CA@

Ici, le premier terme du deuxième membre est nul. Le deuxième terme, égal à ( I M B / p 2 ) ( S y 2 / m ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fpC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaada WcgaqaaiaadMeacaWGnbGaamOqaaqaaiaadchadaahaaWcbeqaaiaa ikdaaaaaaaGccaGLOaGaayzkaaWaaeWaaeaadaWcgaqaaiaadofada qhaaWcbaGaamyEaaqaaiaaikdaaaaakeaacaWGTbaaaaGaayjkaiaa wMcaaaaa@3F66@ avec I M B MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fpC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysaiaad2 eacaWGcbaaaa@3685@ donné en (3.3), devient une deuxième approximation pour le terme de pénalisation en (A.1). Par conséquent, S Δ 2 ( 1 / m 1 / n ) S y 2 + ( I M B / p 2 ) ( S y 2 / m ) . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fpC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uamaaDa aaleaacqqHuoaraeaacaaIYaaaaOGaeyisIS7aaeWaaeaadaWcgaqa aiaaigdaaeaacaWGTbaaaiabgkHiTmaalyaabaGaaGymaaqaaiaad6 gaaaaacaGLOaGaayzkaaGaam4uamaaDaaaleaacaWG5baabaGaaGOm aaaakiabgUcaRmaabmaabaWaaSGbaeaacaWGjbGaamytaiaadkeaae aacaWGWbWaaWbaaSqabeaacaaIYaaaaaaaaOGaayjkaiaawMcaamaa bmaabaWaaSGbaeaacaWGtbWaa0baaSqaaiaadMhaaeaacaaIYaaaaa GcbaGaamyBaaaaaiaawIcacaGLPaaacaGGUaaaaa@4EA2@ Cela donne le résultat souhaité (7.4).

Annexe 2

Comparaison de deux formes quadratiques

Nous comparons les deux formes quadratiques en x ¯ r x ¯ s , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fpC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabCiEayaara WaaSbaaSqaaiaadkhaaeqaaOGaeyOeI0IabCiEayaaraWaaSbaaSqa aiaadohaaeqaaOGaaiilaaaa@3A48@ Q r MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fpC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcaaQaamyuaO WaaSbaaKqaGgaacaWGYbaabeaaaaa@3769@ et Q s MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fpC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcaaQaamyuaO WaaSbaaKqaGgaacaWGZbaabeaaaaa@376A@ définies en (3.1), et justifions l’approximation Q r Q s MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fpC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcaaQaamyuaO WaaSbaaSqaaiaadkhaaeqaaOGaeyisISBcaaQaamyuaOWaaSbaaKqa GgaacaWGZbaabeaaaaa@3BD1@ nécessaire dans la preuve à l’annexe 3 du résultat 2. Les matrices de pondération respectives, Σ r MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fpC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaC4OdmaaBa aaleaacaWGYbaabeaaaaa@3670@ et Σ s , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fpC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaC4OdmaaBa aaleaacaWGZbaabeaakiaacYcaaaa@372B@ sont définies positives. Par conséquent, Q r MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fpC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcaaQaamyuaO WaaSbaaKqaGgaacaWGYbaabeaaaaa@3769@ (ou Q s ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fpC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeGaaeaaja aOcaWGrbGcdaWgaaqcbaAaaiaadohaaeqaaaGccaGLPaaaaaa@383C@ ne peut être égal à zéro que sous l’équilibre parfait x ¯ r = x ¯ s . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fpC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabCiEayaara WaaSbaaSqaaiaadkhaaeqaaOGaeyypa0JabCiEayaaraWaaSbaaSqa aiaadohaaeqaaOGaaiOlaaaa@3A63@ Puisque Q r = Q s MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fpC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyuamaaBa aaleaacaWGYbaabeaakiabg2da9iaadgfadaWgaaWcbaGaam4Caaqa baaaaa@3921@ pour l’équilibre parfait, l’argument de continuité implique que Q r Q s MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fpC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyuamaaBa aaleaacaWGYbaabeaakiabgIKi7kaadgfadaWgaaWcbaGaam4Caaqa baaaaa@39CC@ pour des ensembles de réponses quasi équilibrés. Dans quelle mesure sont-ils proches plus généralement ?

L’estimateur CAL (5.1) utilise les facteurs de pondération g k = x ¯ s Σ r 1 x k , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fpC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcaaQaam4zaO WaaSbaaKqaGgaacaWGRbaabeaajaaOcqGH9aqpceWH4bGbaeHbauaa kmaaBaaajeaObaGaam4CaaqabaGccaWHJoWaa0baaSqaaiaadkhaae aacqGHsislcaaIXaaaaOGaaCiEamaaBaaaleaacaWGRbaabeaakiaa cYcaaaa@42FE@ définis pour tout k s . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fpC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4AaiabgI GiolaadohacaGGUaaaaa@383C@ Leur lien avec Q r MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fpC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyuamaaBa aaleaacaWGYbaabeaaaaa@3617@ est montré dans les deuxième et troisième expressions en (A.3) ci-dessous. Considérons également les facteurs f k = x ¯ r Σ s 1 x k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fpC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcaaQaamOzaO WaaSbaaKqaGgaacaWGRbaabeaajaaOcqGH9aqpkiqahIhagaqegaqb amaaBaaaleaacaWGYbaabeaakiaaho6adaqhaaWcbaGaam4Caaqaai abgkHiTiaaigdaaaGccaWH4bWaaSbaaSqaaiaadUgaaeqaaaaa@41A4@ pour k s . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fpC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4AaiabgI GiolaadohacaGGUaaaaa@383C@ Ils jouent un rôle déterminant pour Q s , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fpC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyuamaaBa aaleaacaWGZbaabeaakiaacYcaaaa@36D2@ et pour I M B = P 2 Q s , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fpC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysaiaad2 eacaWGcbGaeyypa0JaamiuamaaCaaaleqabaGaaGOmaaaakiaadgfa daWgaaWcbaGaam4CaaqabaGccaGGSaaaaa@3C07@ comme le montrent les deux dernières expressions dans (A.3). Les moments qui suivent de g k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fpC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4zamaaBa aaleaacaWGRbaabeaaaaa@3626@ et f k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fpC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcaaQaamOzaO WaaSbaaKqaGgaacaWGRbaabeaaaaa@3777@ sont vérifiés à l’aide de la condition (2.2) appliquée au vecteur x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fpC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCiEaaaa@351F@ :

g ¯ r = 1 , var r ( g ) = Q r , g ¯ s = 1 + Q r ; f ¯ s = 1 , var s ( f ) = Q s , f ¯ r = 1 + Q s . ( A .3 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fpC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabm4zayaara WaaSbaaSqaaiaadkhaaeqaaOGaeyypa0JaaGymaiaacYcaciGG2bGa aiyyaiaackhadaWgaaWcbaGaamOCaaqabaGcdaqadaqaaiaadEgaai aawIcacaGLPaaacqGH9aqpcaWGrbWaaSbaaSqaaiaadkhaaeqaaOGa aiilaiaaysW7caaMc8Uabm4zayaaraWaaSbaaSqaaiaadohaaeqaaO Gaeyypa0JaaGymaiabgUcaRiaadgfadaWgaaWcbaGaamOCaaqabaGc caGG7aGaaGzbVlqadAgagaqeamaaBaaaleaacaWGZbaabeaakiabg2 da9iaaigdacaGGSaGaciODaiaacggacaGGYbWaaSbaaSqaaiaadoha aeqaaOWaaeWaaeaacaWGMbaacaGLOaGaayzkaaGaeyypa0Jaamyuam aaBaaaleaacaWGZbaabeaakiaacYcacaaMe8UaaGPaVlqadAgagaqe amaaBaaaleaacaWGYbaabeaakiabg2da9iaaigdacqGHRaWkcaWGrb WaaSbaaSqaaiaadohaaeqaaOGaaiOlaiaaywW7caaMf8UaaGzbVlaa ywW7caGGOaGaaeyqaiaab6cacaqGZaGaaiykaaaa@7260@

Pour g k , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fpC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4zamaaBa aaleaacaWGRbaabeaakiaacYcaaaa@36E0@ les moyennes sont définies comme étant g ¯ s = s d k g k / s d k , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fpC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabm4zayaara WaaSbaaSqaaiaadohaaeqaaOGaeyypa0ZaaSGbaeaadaaeqaqaaiaa dsgadaWgaaWcbaGaam4AaaqabaGccaWGNbWaaSbaaSqaaiaadUgaae qaaaqaaiaadohaaeqaniabggHiLdaakeaadaaeqaqaaiaadsgadaWg aaWcbaGaam4AaaqabaaabaGaam4Caaqab0GaeyyeIuoaaaGccaGGSa aaaa@43EC@ g ¯ r = r d k g k / r d k , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fpC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabm4zayaara WaaSbaaSqaaiaadkhaaeqaaOGaeyypa0ZaaSGbaeaadaaeqaqaaiaa dsgadaWgaaWcbaGaam4AaaqabaGccaWGNbWaaSbaaSqaaiaadUgaae qaaaqaaiaadkhaaeqaniabggHiLdaakeaadaaeqaqaaiaadsgadaWg aaWcbaGaam4AaaqabaaabaGaamOCaaqab0GaeyyeIuoaaaGccaGGSa aaaa@43E9@ et les variances sont var s ( g ) = s d k ( g k g ¯ s ) 2 / s d k , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fpC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcaaMaciODai aacggacaGGYbGcdaWgaaqcbawaaiaadohaaeqaaOWaaeWaaeaacaWG NbaacaGLOaGaayzkaaqcaaMaeyypa0JcdaWcgaqaamaaqabajaayba GaamizaOWaaSbaaKqaGfaacaWGRbaabeaakmaabmaabaGaam4zamaa BaaaleaacaWGRbaabeaakiabgkHiTiqadEgagaqeamaaBaaaleaaca WGZbaabeaaaOGaayjkaiaawMcaamaaCaaajeaybeqaaiaaikdaaaaa baGaam4CaaqabKWaGjabggHiLdaakeaadaaeqaqcaawaaiaadsgakm aaBaaajeaybaGaam4AaaqabaaabaGaam4CaaqabKWaGjabggHiLdaa aOGaaiilaaaa@51D5@ var r ( g ) = r d k ( g k g ¯ r ) 2 / r d k . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fpC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciODaiaacg gacaGGYbWaaSbaaSqaaiaadkhaaeqaaOWaaeWaaeaacaWGNbaacaGL OaGaayzkaaGaeyypa0ZaaSGbaeaadaaeqaqaaiaadsgadaWgaaWcba Gaam4AaaqabaGcdaqadaqaaiaadEgadaWgaaWcbaGaam4AaaqabaGc cqGHsislceWGNbGbaebadaWgaaWcbaGaamOCaaqabaaakiaawIcaca GLPaaadaahaaWcbeqaaiaaikdaaaaabaGaamOCaaqab0GaeyyeIuoa aOqaamaaqababaGaamizamaaBaaaleaacaWGRbaabeaaaeaacaWGYb aabeqdcqGHris5aaaakiaac6caaaa@4DCD@ Pour les moments correspondants de f k , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fpC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaaBa aaleaacaWGRbaabeaakiaacYcaaaa@36DF@ remplaçons g k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fpC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4zamaaBa aaleaacaWGRbaabeaaaaa@3626@ par f k . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fpC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaaBa aaleaacaWGRbaabeaakiaac6caaaa@36E1@ Les variances var s ( g ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fpC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciODaiaacg gacaGGYbWaaSbaaSqaaiaadohaaeqaaOWaaeWaaeaacaWGNbaacaGL OaGaayzkaaaaaa@3A98@ et var r ( f ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fpC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciODaiaacg gacaGGYbWaaSbaaSqaaiaadkhaaeqaaOWaaeWaaeaacaWGMbaacaGL OaGaayzkaaaaaa@3A96@ n’ont pas une forme aussi transparente et seront approximées. Une autre propriété importante découlant de (2.2) est s d k f k g k / s d k = r d k f k g k / r d k = 1 . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fpC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSGbaeaada aeqaqaaiaadsgadaWgaaWcbaGaam4AaaqabaGccaWGMbWaaSbaaSqa aiaadUgaaeqaaOGaam4zamaaBaaaleaacaWGRbaabeaaaeaacaWGZb aabeqdcqGHris5aaGcbaWaaabeaeaacaWGKbWaaSbaaSqaaiaadUga aeqaaaqaaiaadohaaeqaniabggHiLdaaaOGaeyypa0ZaaSGbaeaada aeqaqaaiaadsgadaWgaaWcbaGaam4AaaqabaGccaWGMbWaaSbaaSqa aiaadUgaaeqaaOGaam4zamaaBaaaleaacaWGRbaabeaaaeaacaWGYb aabeqdcqGHris5aaGcbaWaaabeaeaacaWGKbWaaSbaaSqaaiaadUga aeqaaaqaaiaadkhaaeqaniabggHiLdGccqGH9aqpcaaIXaaaaiaac6 caaaa@5383@ Ces équations et expressions appropriées dans (A.3) donnent

cov s ( f , g ) = s d k ( f k f ¯ s ) ( g k g ¯ s ) / s d k = 1 f ¯ s g ¯ s = Q r , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fpC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4yaiaac+ gacaGG2bWaaSbaaSqaaiaadohaaeqaaOWaaeWaaeaacaWGMbGaaiil aiaadEgaaiaawIcacaGLPaaacqGH9aqpdaWcgaqaamaaqababaGaam izamaaBaaaleaacaWGRbaabeaakmaabmaabaGaamOzamaaBaaaleaa caWGRbaabeaakiabgkHiTiqadAgagaqeamaaBaaaleaacaWGZbaabe aaaOGaayjkaiaawMcaamaabmaabaGaam4zamaaBaaaleaacaWGRbaa beaakiabgkHiTiqadEgagaqeamaaBaaaleaacaWGZbaabeaaaOGaay jkaiaawMcaaaWcbaGaam4Caaqab0GaeyyeIuoaaOqaamaaqababaGa amizamaaBaaaleaacaWGRbaabeaaaeaacaWGZbaabeqdcqGHris5aa aakiabg2da9iaaigdacqGHsislceWGMbGbaebadaWgaaWcbaGaam4C aaqabaGccaaMc8Uabm4zayaaraWaaSbaaSqaaiaadohaaeqaaOGaey ypa0JaeyOeI0IaamyuamaaBaaaleaacaWGYbaabeaakiaacYcaaaa@61D4@

cov r ( f , g ) = r d k ( f k f ¯ r ) ( g k g ¯ r ) / r d k = 1 f ¯ r g ¯ r = Q s . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fpC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4yaiaac+ gacaGG2bWaaSbaaSqaaiaadkhaaeqaaOWaaeWaaeaacaWGMbGaaiil aiaadEgaaiaawIcacaGLPaaacqGH9aqpdaWcgaqaamaaqababaGaam izamaaBaaaleaacaWGRbaabeaakmaabmaabaGaamOzamaaBaaaleaa caWGRbaabeaakiabgkHiTiqadAgagaqeamaaBaaaleaacaWGYbaabe aaaOGaayjkaiaawMcaamaabmaabaGaam4zamaaBaaaleaacaWGRbaa beaakiabgkHiTiqadEgagaqeamaaBaaaleaacaWGYbaabeaaaOGaay jkaiaawMcaaaWcbaGaamOCaaqab0GaeyyeIuoaaOqaamaaqababaGa amizamaaBaaaleaacaWGRbaabeaaaeaacaWGYbaabeqdcqGHris5aa aakiabg2da9iaaigdacqGHsislceWGMbGbaebadaWgaaWcbaGaamOC aaqabaGccaaMc8Uabm4zayaaraWaaSbaaSqaaiaadkhaaeqaaOGaey ypa0JaeyOeI0IaamyuamaaBaaaleaacaWGZbaabeaakiaac6caaaa@61D0@

Maintenant, utilisons cov s 2 ( f , g ) var s ( f ) var s ( g ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fpC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4yaiaac+ gacaGG2bWaa0baaSqaaiaadohaaeaacaaIYaaaaOWaaeWaaeaacaWG MbGaaiilaiaadEgaaiaawIcacaGLPaaacqGHKjYOciGG2bGaaiyyai aackhadaWgaaWcbaGaam4CaaqabaGcdaqadaqaaiaadAgaaiaawIca caGLPaaaciGG2bGaaiyyaiaackhadaWgaaWcbaGaam4CaaqabaGcda qadaqaaiaadEgaaiaawIcacaGLPaaaaaa@4B97@ et l’inégalité analogue où r MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fpC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCaaaa@3515@ remplace s . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fpC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4Caiaac6 caaaa@35C8@ En utilisant aussi var s ( f ) = Q s MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fpC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciODaiaacg gacaGGYbWaaSbaaSqaaiaadohaaeqaaOWaaeWaaeaacaWGMbaacaGL OaGaayzkaaGaeyypa0JaamyuamaaBaaaleaacaWGZbaabeaaaaa@3D97@ et var r ( g ) = Q r MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fpC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciODaiaacg gacaGGYbWaaSbaaSqaaiaadkhaaeqaaOWaaeWaaeaacaWGNbaacaGL OaGaayzkaaGaeyypa0JaamyuamaaBaaaleaacaWGYbaabeaaaaa@3D96@ provenant de (A.3), nous obtenons les bornes pour le ratio Q r / Q s : MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fpC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSGbaeaaca WGrbWaaSbaaSqaaiaadkhaaeqaaaGcbaGaamyuamaaBaaaleaacaWG ZbaabeaaaaGccaaMe8UaaiOoaaaa@3A86@

Q s var r ( f ) Q r Q s var s ( g ) Q r . ( A .4 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fpC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca WGrbWaaSbaaSqaaiaadohaaeqaaaGcbaGaciODaiaacggacaGGYbWa aSbaaSqaaiaadkhaaeqaaOWaaeWaaeaacaWGMbaacaGLOaGaayzkaa aaaiabgsMiJoaalaaabaGaamyuamaaBaaaleaacaWGYbaabeaaaOqa aiaadgfadaWgaaWcbaGaam4CaaqabaaaaOGaeyizIm6aaSaaaeaaci GG2bGaaiyyaiaackhadaWgaaWcbaGaam4CaaqabaGcdaqadaqaaiaa dEgaaiaawIcacaGLPaaaaeaacaWGrbWaaSbaaSqaaiaadkhaaeqaaa aakiaac6cacaaMf8UaaGzbVlaaywW7caaMf8UaaGzbVlaacIcacaqG bbGaaeOlaiaabsdacaGGPaaaaa@58B4@

Afin d’obtenir des bornes supérieure et inférieure plus transparentes, approximons les deux variances en (A.4) en supposant que le coefficient de variation (écart-type divisé par la moyenne) est approximativement le même pour la réponse r MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fpC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCaaaa@3515@ que pour l’échantillon s , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fpC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4CaiaacY caaaa@35C6@ et cela pour f MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fpC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaaaa@3509@ ainsi que g . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fpC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4zaiaac6 caaaa@35BC@ Cela suppose une certaine stabilité du coefficient de variation. Alors, var s ( g ) ( g ¯ s ) 2 var r ( g ) / ( g ¯ r ) 2 = ( 1 + Q r ) 2 Q r , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fpC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSGbaeaaci GG2bGaaiyyaiaackhadaWgaaWcbaGaam4CaaqabaGcdaqadaqaaiaa dEgaaiaawIcacaGLPaaacqGHijYUdaqadaqaaiqadEgagaqeamaaBa aaleaacaWGZbaabeaaaOGaayjkaiaawMcaamaaCaaaleqabaGaaGOm aaaakiGacAhacaGGHbGaaiOCamaaBaaaleaacaWGYbaabeaakmaabm aabaGaam4zaaGaayjkaiaawMcaaaqaamaabmaabaGabm4zayaaraWa aSbaaSqaaiaadkhaaeqaaaGccaGLOaGaayzkaaWaaWbaaSqabeaaca aIYaaaaOGaeyypa0ZaaeWaaeaacaaIXaGaey4kaSIaamyuamaaBaaa leaacaWGYbaabeaaaOGaayjkaiaawMcaamaaCaaaleqabaGaaGOmaa aakiaadgfadaWgaaWcbaGaamOCaaqabaaaaOGaaiilaaaa@5608@ de sorte que la borne supérieure en (A.4) est approximativement ( 1 + Q r ) 2 > 1. MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fpC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaaca aIXaGaey4kaSIaamyuamaaBaaaleaacaWGYbaabeaaaOGaayjkaiaa wMcaamaaCaaaleqabaGaaGOmaaaakiabg6da+iaaigdacaGGUaaaaa@3CAF@ De même, var r ( f ) ( f ¯ r ) 2 var s ( f ) / ( f ¯ s ) 2 = ( 1 + Q s ) 2 Q s , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fpC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSGbaeaaci GG2bGaaiyyaiaackhadaWgaaWcbaGaamOCaaqabaGcdaqadaqaaiaa dAgaaiaawIcacaGLPaaacqGHijYUdaqadaqaaiqadAgagaqeamaaBa aaleaacaWGYbaabeaaaOGaayjkaiaawMcaamaaCaaaleqabaGaaGOm aaaakiGacAhacaGGHbGaaiOCamaaBaaaleaacaWGZbaabeaakmaabm aabaGaamOzaaGaayjkaiaawMcaaaqaamaabmaabaGabmOzayaaraWa aSbaaSqaaiaadohaaeqaaaGccaGLOaGaayzkaaWaaWbaaSqabeaaca aIYaaaaOGaeyypa0ZaaeWaaeaacaaIXaGaey4kaSIaamyuamaaBaaa leaacaWGZbaabeaaaOGaayjkaiaawMcaamaaCaaaleqabaGaaGOmaa aakiaadgfadaWgaaWcbaGaam4CaaqabaaaaOGaaiilaaaa@5606@ ce qui donne ( 1 + Q s ) 2 < 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fpC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaaca aIXaGaey4kaSIaamyuamaaBaaaleaacaWGZbaabeaaaOGaayjkaiaa wMcaamaaCaaaleqabaGaeyOeI0IaaGOmaaaakiabgYda8iaaigdaaa a@3CE7@ comme borne inférieure approximative en (A.4). L’intervalle approximatif pour le ratio Q r / Q s MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fpC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSGbaeaaca WGrbWaaSbaaSqaaiaadkhaaeqaaaGcbaGaamyuamaaBaaaleaacaWG Zbaabeaaaaaaaa@3831@ est donc

Q r / Q s ( ( 1 + Q s ) 2 , ( 1 + Q r ) 2 ) . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fpC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSGbaeaaca WGrbWaaSbaaSqaaiaadkhaaeqaaaGcbaGaamyuamaaBaaaleaacaWG ZbaabeaaaaGccqGHiiIZdaqadaqaamaabmaabaGaaGymaiabgUcaRi aadgfadaWgaaWcbaGaam4CaaqabaaakiaawIcacaGLPaaadaahaaWc beqaaiabgkHiTiaaikdaaaGccaGGSaWaaeWaaeaacaaIXaGaey4kaS IaamyuamaaBaaaleaacaWGYbaabeaaaOGaayjkaiaawMcaamaaCaaa leqabaGaaGOmaaaaaOGaayjkaiaawMcaaiaac6caaaa@49D0@

Cela illustre le fait que le ratio n’est pas très éloigné de 1, car pour la plupart des données, les valeurs de Q s MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fpC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcaaQaamyuaO WaaSbaaKqaGgaacaWGZbaabeaaaaa@376A@ et Q r MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fpC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcaaQaamyuaO WaaSbaaKqaGgaacaWGYbaabeaaaaa@3769@ sont toutes deux faibles comparativement à 1, Q r MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fpC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcaaQaamyuaO WaaSbaaKqaGgaacaWGYbaabeaaaaa@3769@ étant habituellement la valeur quelque peu plus grande. Des travaux empiriques donnent néanmoins à penser que la borne supérieure approximative ( 1 + Q r ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fpC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaaca aIXaGaey4kaSIaamyuamaaBaaaleaacaWGYbaabeaaaOGaayjkaiaa wMcaamaaCaaaleqabaGaaGOmaaaaaaa@3A30@ peut souvent être trop faible.

Annexe 3

Obtention du résultat 2

Nous calculons les expressions en (8.2) sous les conditions énoncées. Les tailles de r MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fpC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCaaaa@3515@ et s MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fpC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4Caaaa@3516@ sont m MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fpC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBaaaa@3510@ et n , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fpC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaiaacY caaaa@35C1@ respectivement; le taux de réponse est p = m / n . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fpC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCaiabg2 da9maalyaabaGaamyBaaqaaiaad6gaaaGaaiOlaaaa@38C6@ L’écart de l’estimateur CAL par rapport à l’estimateur sans biais FUL est Y ^ C A L Y ^ F U L = N ^ Δ r MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fpC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmywayaaja WaaSbaaSqaaiaadoeacaWGbbGaamitaaqabaGccqGHsislceWGzbGb aKaadaWgaaWcbaGaamOraiaadwfacaWGmbaabeaakiabg2da9iqad6 eagaqcaiabfs5aenaaBaaaleaacaWGYbaabeaaaaa@409A@

Δ r = ( b r b s ) x ¯ s = Σ r d k g k y k / Σ r d k Σ s d k y k / Σ s d k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fpC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuiLdq0aaS baaSqaaiaadkhaaeqaaOGaeyypa0ZaaeWaaeaacaWHIbWaaSbaaSqa aiaadkhaaeqaaOGaeyOeI0IaaCOyamaaBaaaleaacaWGZbaabeaaaO GaayjkaiaawMcaamaaCaaaleqabaGccWaGGBOmGikaaiqahIhagaqe amaaBaaaleaacaWGZbaabeaakiabg2da9maalyaabaGaeu4Odm1aaS baaSqaaiaadkhaaeqaaOGaamizamaaBaaaleaacaWGRbaabeaakiaa dEgadaWgaaWcbaGaam4AaaqabaGccaWG5bWaaSbaaSqaaiaadUgaae qaaaGcbaGaeu4Odm1aaSbaaSqaaiaadkhaaeqaaOGaamizamaaBaaa leaacaWGRbaabeaaaaGccqGHsisldaWcgaqaaiabfo6atnaaBaaale aacaWGZbaabeaakiaadsgadaWgaaWcbaGaam4AaaqabaGccaWG5bWa aSbaaSqaaiaadUgaaeqaaaGcbaGaeu4Odm1aaSbaaSqaaiaadohaae qaaOGaamizamaaBaaaleaacaWGRbaabeaaaaaaaa@5F35@

avec b r MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fpC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCOyamaaBa aaleaacaWGYbaabeaaaaa@362C@ et b s MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fpC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCOyamaaBa aaleaacaWGZbaabeaaaaa@362D@ donnés par (4.1), et g k = x ¯ s Σ r 1 x k . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fpC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4zamaaBa aaleaacaWGRbaabeaakiabg2da9iqahIhagaqegaqbamaaBaaaleaa caWGZbaabeaakiaaho6adaqhaaWcbaGaamOCaaqaaiabgkHiTiaaig daaaGccaWH4bWaaSbaaSqaaiaadUgaaeqaaOGaaiOlaaaa@4066@ Notons que b s x ¯ s = y ¯ s MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fpC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabCOyayaafa WaaSbaaSqaaiaadohaaeqaaOGabCiEayaaraWaaSbaaSqaaiaadoha aeqaaOGaeyypa0JabmyEayaaraWaaSbaaSqaaiaadohaaeqaaaaa@3BCA@ en vertu de (2.2). Or, Σ r d k g k x k / Σ r d k = Σ s d k x k / Σ s d k = x ¯ s . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fpC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSGbaeaacq qHJoWudaWgaaWcbaGaamOCaaqabaGccaWGKbWaaSbaaSqaaiaadUga aeqaaOGaam4zamaaBaaaleaacaWGRbaabeaakiqahIhagaqbamaaBa aaleaacaWGRbaabeaaaOqaaiabfo6atnaaBaaaleaacaWGYbaabeaa kiaadsgadaWgaaWcbaGaam4AaaqabaaaaOGaeyypa0ZaaSGbaeaacq qHJoWudaWgaaWcbaGaam4CaaqabaGccaWGKbWaaSbaaSqaaiaadUga aeqaaOGabCiEayaafaWaaSbaaSqaaiaadUgaaeqaaaGcbaGaeu4Odm 1aaSbaaSqaaiaadohaaeqaaOGaamizamaaBaaaleaacaWGRbaabeaa aaGccqGH9aqpceWH4bGbaeHbauaadaWgaaWcbaGaam4CaaqabaGcca GGUaaaaa@52D4@ En utilisant le résultat de la multiplication à droite de cette équation par β , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fpC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCOSdiaacY caaaa@360C@ nous obtenons Δ r = Σ r d k g k ( y k x k β ) / Σ r d k Σ s d k ( y k x k β ) / Σ s d k , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fpC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuiLdq0aaS baaSqaaiaadkhaaeqaaOGaeyypa0ZaaSGbaeaacqqHJoWudaWgaaWc baGaamOCaaqabaGccaWGKbWaaSbaaSqaaiaadUgaaeqaaOGaam4zam aaBaaaleaacaWGRbaabeaakmaabmaabaGaamyEamaaBaaaleaacaWG RbaabeaakiabgkHiTiqahIhagaqbamaaBaaaleaacaWGRbaabeaaki aahk7aaiaawIcacaGLPaaaaeaacqqHJoWudaWgaaWcbaGaamOCaaqa baGccaWGKbWaaSbaaSqaaiaadUgaaeqaaaaakiabgkHiTmaalyaaba Gaeu4Odm1aaSbaaSqaaiaadohaaeqaaOGaamizamaaBaaaleaacaWG RbaabeaakmaabmaabaGaamyEamaaBaaaleaacaWGRbaabeaakiabgk HiTiqahIhagaqbamaaBaaaleaacaWGRbaabeaakiaahk7aaiaawIca caGLPaaaaeaacqqHJoWudaWgaaWcbaGaam4CaaqabaGccaWGKbWaaS baaSqaaiaadUgaaeqaaaaakiaacYcaaaa@5EAA@ qui exprime Δ r MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fpC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuiLdq0aaS baaSqaaiaadkhaaeqaaaaa@36A7@ en fonction des résidus ε k = y k x k β MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fpC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcaaQaeqyTdu McdaWgaaqcbaAaaiaadUgaaeqaaKaaGkabg2da9iaadMhakmaaBaaa jeaObaGaam4AaaqabaqcaaQaeyOeI0IabCiEayaafaGcdaWgaaqcba AaaiaadUgaaeqaaKaaGkaahk7aaaa@42F4@ du modèle (8.1):

Δ r = r d k g k ε k r d k s d k ε k s d k . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fpC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuiLdq0aaS baaSqaaiaadkhaaeqaaOGaeyypa0ZaaSaaaeaadaaeqaqaaiaadsga daWgaaWcbaGaam4AaaqabaGccaWGNbWaaSbaaSqaaiaadUgaaeqaaO GaeqyTdu2aaSbaaSqaaiaadUgaaeqaaaqaaiaadkhaaeqaniabggHi LdaakeaadaaeqaqaaiaadsgadaWgaaWcbaGaam4AaaqabaaabaGaam OCaaqab0GaeyyeIuoaaaGccqGHsisldaWcaaqaamaaqababaGaamiz amaaBaaaleaacaWGRbaabeaakiabew7aLnaaBaaaleaacaWGRbaabe aaaeaacaWGZbaabeqdcqGHris5aaGcbaWaaabeaeaacaWGKbWaaSba aSqaaiaadUgaaeqaaaqaaiaadohaaeqaniabggHiLdaaaOGaaiOlaa aa@549B@

Puis, nous utilisons les propriétés sous le modèle de ε k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fpC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcaaQaeqyTdu McdaWgaaqcbaAaaiaadUgaaeqaaaaa@3833@ dans (8.1). De E ξ ( ε k | x k ) = 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fpC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyramaaBa aaleaacqaH+oaEaeqaaOWaaeWaaeaadaabcaqaaiabew7aLnaaBaaa leaacaWGRbaabeaakiaaykW7aiaawIa7aiaaysW7caWH4bWaaSbaaS qaaiaadUgaaeqaaaGccaGLOaGaayzkaaGaeyypa0JaaGimaaaa@43CC@ pour tout k , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fpC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4AaiaacY caaaa@35BE@ il découle que E ξ ( Δ r | X , r , s ) = 0. MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fpC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyramaaBa aaleaacqaH+oaEaeqaaOWaaeWaaeaadaabcaqaaiabfs5aenaaBaaa leaacaWGYbaabeaakiaaykW7aiaawIa7aiaaysW7caWHybGaaiilai aadkhacaGGSaGaam4CaaGaayjkaiaawMcaaiabg2da9iaaicdacaGG Uaaaaa@464D@ Pour évaluer la variance, nous utilisons E ξ ( ε k 2 | x k ) = σ ε 2 , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fpC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyramaaBa aaleaacqaH+oaEaeqaaOWaaeWaaeaadaabcaqaaiabew7aLnaaDaaa leaacaWGRbaabaGaaGOmaaaakiaaykW7aiaawIa7aiaaysW7caWH4b WaaSbaaSqaaiaadUgaaeqaaaGccaGLOaGaayzkaaGaeyypa0Jaeq4W dm3aa0baaSqaaiabew7aLbqaaiaaikdaaaGccaGGSaaaaa@48DC@ pour tout k s , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fpC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4AaiabgI GiolaadohacaGGSaaaaa@383A@ et E ξ ( ε k ε l | x k , x l ) = 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fpC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyramaaBa aaleaacqaH+oaEaeqaaOWaaeWaaeaadaabcaqaaiabew7aLnaaBaaa leaacaWGRbaabeaakiabew7aLnaaBaaaleaacqWItecBaeqaaOGaaG PaVdGaayjcSdGaaGjbVlaahIhadaWgaaWcbaGaam4AaaqabaGccaGG SaGaaCiEamaaBaaaleaacqWItecBaeqaaaGccaGLOaGaayzkaaGaey ypa0JaaGimaaaa@49F2@ pour tout k l s . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fpC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4Aaiabgc Mi5kabloriSjabgIGiolaadohacaGGUaaaaa@3B34@ Cela donne

E ξ ( Δ r 2 | X , r , s ) = σ ε 2 r d k 2 g k 2 ( r d k ) 2 + σ ε 2 s d k 2 ( s d k ) 2 2 σ ε 2 r d k 2 g k ( r d k ) ( s d k ) . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fpC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyramaaBa aaleaacqaH+oaEaeqaaOWaaeWaaeaadaabcaqaaiabfs5aenaaDaaa leaacaWGYbaabaGaaGOmaaaakiaaykW7aiaawIa7aiaaysW7caWHyb GaaiilaiaadkhacaGGSaGaam4CaaGaayjkaiaawMcaaiabg2da9iab eo8aZnaaDaaaleaacqaH1oqzaeaacaaIYaaaaOWaaSaaaeaadaaeqa qaaiaadsgadaqhaaWcbaGaam4AaaqaaiaaikdaaaGccaWGNbWaa0ba aSqaaiaadUgaaeaacaaIYaaaaaqaaiaadkhaaeqaniabggHiLdaake aadaqadaqaamaaqababaGaamizamaaBaaaleaacaWGRbaabeaaaeaa caWGYbaabeqdcqGHris5aaGccaGLOaGaayzkaaWaaWbaaSqabeaaca aIYaaaaaaakiabgUcaRiabeo8aZnaaDaaaleaacqaH1oqzaeaacaaI YaaaaOWaaSaaaeaadaaeqaqaaiaadsgadaqhaaWcbaGaam4Aaaqaai aaikdaaaaabaGaam4Caaqab0GaeyyeIuoaaOqaamaabmaabaWaaabe aeaacaWGKbWaaSbaaSqaaiaadUgaaeqaaaqaaiaadohaaeqaniabgg HiLdaakiaawIcacaGLPaaadaahaaWcbeqaaiaaikdaaaaaaOGaeyOe I0IaaGOmaiabeo8aZnaaDaaaleaacqaH1oqzaeaacaaIYaaaaOWaaS aaaeaadaaeqaqaaiaadsgadaqhaaWcbaGaam4AaaqaaiaaikdaaaGc caWGNbWaaSbaaSqaaiaadUgaaeqaaaqaaiaadkhaaeqaniabggHiLd aakeaadaqadaqaamaaqababaGaamizamaaBaaaleaacaWGRbaabeaa aeaacaWGYbaabeqdcqGHris5aaGccaGLOaGaayzkaaWaaeWaaeaada aeqaqaaiaadsgadaWgaaWcbaGaam4AaaqabaaabaGaam4Caaqab0Ga eyyeIuoaaOGaayjkaiaawMcaaaaacaGGUaaaaa@8758@

Ici, d k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fpC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamizamaaBa aaleaacaWGRbaabeaaaaa@3623@ s’élimine, parce que sa valeur est constante. Les première et deuxième expressions dans (A.3) sont vérifiées pour tout d k , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fpC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamizamaaBa aaleaacaWGRbaabeaakiaacYcaaaa@36DD@ en particulier d k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fpC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamizamaaBa aaleaacaWGRbaabeaaaaa@3623@ constant, de sorte que nous obtenons r g k / m = 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fpC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSGbaeaada aeqaqaaiaadEgadaWgaaWcbaGaam4AaaqabaaabaGaamOCaaqab0Ga eyyeIuoaaOqaaiaad2gacqGH9aqpcaaIXaaaaaaa@3BC8@ pour la moyenne et r g k 2 / m = Q r + 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fpC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSGbaeaada aeqaqaaiaadEgadaqhaaWcbaGaam4AaaqaaiaaikdaaaaabaGaamOC aaqab0GaeyyeIuoaaOqaaiaad2gacqGH9aqpcaWGrbWaaSbaaSqaai aadkhaaeqaaOGaey4kaSIaaGymaaaaaaa@3F6A@ pour la variance plus le carré de la moyenne. Par conséquent,

E ξ ( Δ r 2 | X , r , s ) = ( 1 m ( 1 + Q r ) + 1 n 2 1 n ) σ ε 2 = ( 1 m 1 n + Q r m ) σ ε 2 . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fpC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyramaaBa aaleaacqaH+oaEaeqaaOWaaeWaaeaadaabcaqaaiabfs5aenaaDaaa leaacaWGYbaabaGaaGOmaaaakiaaykW7aiaawIa7aiaaysW7caWHyb GaaiilaiaadkhacaGGSaGaam4CaaGaayjkaiaawMcaaiabg2da9maa bmaabaWaaSaaaeaacaaIXaaabaGaamyBaaaadaqadaqaaiaaigdacq GHRaWkcaWGrbWaaSbaaSqaaiaadkhaaeqaaaGccaGLOaGaayzkaaGa ey4kaSYaaSaaaeaacaaIXaaabaGaamOBaaaacqGHsislcaaIYaWaaS aaaeaacaaIXaaabaGaamOBaaaaaiaawIcacaGLPaaacqaHdpWCdaqh aaWcbaGaeqyTdugabaGaaGOmaaaakiabg2da9maabmaabaWaaSaaae aacaaIXaaabaGaamyBaaaacqGHsisldaWcaaqaaiaaigdaaeaacaWG UbaaaiabgUcaRmaalaaabaGaamyuamaaBaaaleaacaWGYbaabeaaaO qaaiaad2gaaaaacaGLOaGaayzkaaGaeq4Wdm3aa0baaSqaaiabew7a LbqaaiaaikdaaaGccaGGUaaaaa@685D@

En guise d’étape finale, nous utilisons l’approximation Q r Q s MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fpC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyuamaaBa aaleaacaWGYbaabeaakiabgIKi7kaadgfadaWgaaWcbaGaam4Caaqa baaaaa@39CC@ justifiée à l’annexe 2, et I M B = p 2 Q s . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fpC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysaiaad2 eacaWGcbGaeyypa0JaamiCamaaCaaaleqabaGaaGOmaaaakiaadgfa daWgaaWcbaGaam4CaaqabaGccaGGUaaaaa@3C29@ Alors, comme il est affirmé dans le résultat 2, E ξ ( Δ r 2 | X , r , s ) ( 1 p + I M B / p 2 ) ( σ ε 2 / m ) . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fpC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyramaaBa aaleaacqaH+oaEaeqaaOWaaeWaaeaadaabcaqaaiabfs5aenaaDaaa leaacaWGYbaabaGaaGOmaaaakiaaykW7aiaawIa7aiaaysW7caWHyb GaaiilaiaadkhacaGGSaGaam4CaaGaayjkaiaawMcaaiabgIKi7oaa bmaabaGaaGymaiabgkHiTiaadchacqGHRaWkdaWcgaqaaiaadMeaca WGnbGaamOqaaqaaiaadchadaahaaWcbeqaaiaaikdaaaaaaaGccaGL OaGaayzkaaWaaeWaaeaadaWcgaqaaiabeo8aZnaaDaaaleaacqaH1o qzaeaacaaIYaaaaaGcbaGaamyBaaaaaiaawIcacaGLPaaacaGGUaaa aa@5755@

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