Estimation de la variance dans le calage à plusieurs phases
Section 3. Calage avec la distance MCG

Le calage requiert la spécification d’une fonction de distance mesurant la distance entre les poids initiaux et les nouveaux poids calés. Plusieurs fonctions de distance ont été étudiées, certaines étant résumées dans Deville et Särndal (1992). Nous nous concentrons sur la mesure de distance par les moindres carrés généralisée (MCG). La forme classique du calage à plusieurs phases sous la fonction de distance MCG consiste à trouver les valeurs w ˜ i k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabm4Dayaaia WaaSbaaSqaaiaadMgacaWGRbaabeaaaaa@370F@ pour l’ensemble k s i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4AaiabgI GiolaadohadaWgaaWcbaGaamyAaaqabaaaaa@3880@ qui minimisent l’expression

k s i c i k ( w ˜ i k w ˜ i 1 , k w i k ) 2 w ˜ i 1 , k w i k ( 3.1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaabuaeqale aacaWGRbGaeyicI4Saam4CamaaBaaameaacaWGPbaabeaaaSqab0Ga eyyeIuoakmaalaaabaGaam4yamaaBaaaleaacaWGPbGaam4Aaaqaba GcdaqadaqaaiqadEhagaacamaaBaaaleaacaWGPbGaam4AaaqabaGc cqGHsislceWG3bGbaGaadaWgaaWcbaGaamyAaiabgkHiTiaaigdaca aMb8UaaGilaiaaykW7caWGRbaabeaakiaadEhadaWgaaWcbaGaamyA aiaadUgaaeqaaaGccaGLOaGaayzkaaWaaWbaaSqabeaacaaIYaaaaa GcbaGabm4DayaaiaWaaSbaaSqaaiaadMgacqGHsislcaaIXaGaaGza VlaaiYcacaaMc8Uaam4AaaqabaGccaWG3bWaaSbaaSqaaiaadMgaca WGRbaabeaaaaGccaaMf8UaaGzbVlaaywW7caaMf8UaaGzbVlaacIca caaIZaGaaiOlaiaaigdacaGGPaaaaa@66DA@

sous la contrainte

k s i w ˜ i k x i k = k s i 1 w ˜ i 1 , k x i k ( 3.2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaabuaeaace WG3bGbaGaadaWgaaWcbaGaamyAaiaadUgaaeqaaOGaamiEamaaBaaa leaacaWGPbGaam4AaaqabaaabaGaam4AaiabgIGiolaadohadaWgaa adbaGaamyAaaqabaaaleqaniabggHiLdGccaaI9aWaaabuaeaaceWG 3bGbaGaadaWgaaWcbaGaamyAaiabgkHiTiaaigdacaaMb8UaaGilai aaykW7caWGRbaabeaakiaadIhadaWgaaWcbaGaamyAaiaadUgaaeqa aOGaaGzbVlaaywW7caaMf8UaaGzbVlaaywW7caGGOaGaaG4maiaac6 cacaaIYaGaaiykaaWcbaGaam4AaiabgIGiolaadohadaWgaaadbaGa amyAaiabgkHiTiaaigdaaeqaaaWcbeqdcqGHris5aaaa@60E4@

(autrement, on peut écrire w ˜ i 1 , k w i k g i k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabm4Dayaaia WaaSbaaSqaaiaadMgacqGHsislcaaIXaGaaGzaVlaaiYcacaaMc8Ua am4AaaqabaGccaWG3bWaaSbaaSqaaiaadMgacaWGRbaabeaakiaadE gadaWgaaWcbaGaamyAaiaadUgaaeqaaaaa@4292@ au lieu de w ˜ i k ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabm4Dayaaia WaaSbaaSqaaiaadMgacaWGRbaabeaakiaacMcaaaa@37C6@ où les { w ˜ i 1 , k : k s i } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiWaaeaace WG3bGbaGaadaWgaaWcbaGaamyAaiabgkHiTiaaigdacaaMb8UaaGil aiaaykW7caWGRbaabeaakiaaygW7caaI6aGaaGjbVlaadUgacqGHii IZcaWGZbWaaSbaaSqaaiaadMgaaeqaaaGccaGL7bGaayzFaaaaaa@4728@ sont les poids initiaux au début de la phase i , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAaiaacY caaaa@3598@ c’est-à-dire les poids calés obtenus à la phase i 1 ; MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAaiabgk HiTiaaigdacaGG7aaaaa@374F@ les { w ˜ i k : k s i } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiWaaeaace WG3bGbaGaadaWgaaWcbaGaamyAaiaadUgaaeqaaOGaaGzaVlaaiQda caaMe8Uaam4AaiabgIGiolaadohadaWgaaWcbaGaamyAaaqabaaaki aawUhacaGL9baaaaa@41B5@ sont les poids calés de la phase i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAaaaa@34E8@ que nous voulons obtenir; et les { c i k : k s i } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiWaaeaaca WGJbWaaSbaaSqaaiaadMgacaWGRbaabeaakiaaygW7caaI6aGaaGjb VlaadUgacqGHiiIZcaWGZbWaaSbaaSqaaiaadMgaaeqaaaGccaGL7b GaayzFaaaaaa@4192@ sont les facteurs positifs spécifiés utilisés pour contrôler l’importance relative que nous voulons attribuer à chacun des éléments de la somme en fonction de l’information auxiliaire disponible pour k s i 1 . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4AaiabgI GiolaadohadaWgaaWcbaGaamyAaiabgkHiTiaaigdaaeqaaOGaaiOl aaaa@3AE4@ Pour simplifier la notation, supposons à partir de maintenant que c i k = 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4yamaaBa aaleaacaWGPbGaam4AaaqabaGccaaI9aGaaGymaaaa@3878@ pour tout i , k . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAaiaaiY cacaWGRbGaaiOlaaaa@3740@ Les poids résultant de ce scénario de calage sont w ˜ i k = w ˜ i 1, k w i k g i k , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabm4Dayaaia WaaSbaaSqaaiaadMgacaWGRbaabeaakiaai2daceWG3bGbaGaadaWg aaWcbaGaamyAaiabgkHiTiaaigdacaaISaGaam4AaaqabaGccaWG3b WaaSbaaSqaaiaadMgacaWGRbaabeaakiaadEgadaWgaaWcbaGaamyA aiaadUgaaeqaaOGaaiilaaaa@441D@ g i k = 1 + ( l s i 1 w ˜ i 1 , l x i l l s i w ˜ i 1 , l w i l x i l ) T i 1 x i k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4zamaaBa aaleaacaWGPbGaam4AaaqabaGccaaI9aGaaGymaiabgUcaRmaabmaa baWaaabeaeqaleaacaWGSbGaeyicI4Saam4CamaaBaaameaacaWGPb GaeyOeI0IaaGymaaqabaaaleqaniabggHiLdGcceWG3bGbaGaadaWg aaWcbaGaamyAaiabgkHiTiaaigdacaaMb8UaaGilaiaaykW7caWGSb aabeaakiaadIhadaWgaaWcbaGaamyAaiaadYgaaeqaaOGaeyOeI0Ya aabeaeqaleaacaWGSbGaeyicI4Saam4CamaaBaaameaacaWGPbaabe aaaSqab0GaeyyeIuoakiqadEhagaacamaaBaaaleaacaWGPbGaeyOe I0IaaGymaiaaygW7caaISaGaaGPaVlaadYgaaeqaaOGaam4DamaaBa aaleaacaWGPbGaamiBaaqabaGccaWG4bWaaSbaaSqaaiaadMgacaWG SbaabeaaaOGaayjkaiaawMcaamaaCaaaleqabaGccWaGyBOmGikaai aadsfadaqhaaWcbaGaamyAaaqaaiabgkHiTiaaigdaaaGccaWG4bWa aSbaaSqaaiaadMgacaWGRbaabeaaaaa@6E9D@ avec T i = l s i w i l * g i 1, l * x i l x i l . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamivamaaBa aaleaacaWGPbaabeaakiaai2dadaaeqaqabSqaaiaadYgacqGHiiIZ caWGZbWaaSbaaWqaaiaadMgaaeqaaaWcbeqdcqGHris5aOGaam4Dam aaDaaaleaacaWGPbGaamiBaaqaaiaacQcaaaGccaWGNbWaa0baaSqa aiaadMgacqGHsislcaaIXaGaaGilaiaaykW7caWGSbaabaGaaiOkaa aakiaadIhadaWgaaWcbaGaamyAaiaadYgaaeqaaOGaamiEamaaDaaa leaacaWGPbGaamiBaaqaaOGamai2gkdiIcaacaaMb8UaaiOlaaaa@53D9@ D’où, les facteurs de calage dans ce processus agissent multiplicativement pour donner un facteur de calage global g i k * = j = 1 i g j k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4zamaaDa aaleaacaWGPbGaam4AaaqaaiaacQcaaaGccaaI9aWaaebmaeaacaWG NbWaaSbaaSqaaiaadQgacaWGRbaabeaaaeaacaWGQbGaaGypaiaaig daaeaacaWGPbaaniabg+Givdaaaa@40AC@ pour k s i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4AaiabgI GiolaadohadaWgaaWcbaGaamyAaaqabaaaaa@3880@ à la fin de la phase i . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAaiaac6 caaaa@359A@

La mesure de distance (3.1) peut être critiquée, parce que les facteurs 1 / w ˜ i 1, k w i k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSGbaeaaca aIXaaabaGabm4DayaaiaWaaSbaaSqaaiaadMgacqGHsislcaaIXaGa aGilaiaaykW7caWGRbaabeaakiaadEhadaWgaaWcbaGaamyAaiaadU gaaeqaaaaaaaa@3ED9@ pour une phase i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAaaaa@34E8@ pourraient ne pas être forcément tous finis et positifs, car les termes g i 1, k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4zamaaBa aaleaacaWGPbGaeyOeI0IaaGymaiaaiYcacaWGRbaabeaaaaa@394E@ qui figurent dans w ˜ i 1, k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabm4Dayaaia WaaSbaaSqaaiaadMgacqGHsislcaaIXaGaaGilaiaadUgaaeqaaaaa @396D@ au dénominateur peuvent être nuls ou négatifs, ce qui contredit la notion de distance. Un autre choix de fonction de distance, et celui que nous utiliserons dans notre analyse, consiste à remplacer (3.1) par

k s i ( w ˜ i k w ˜ i 1, k w i k ) 2 w i 1, k * w i k ( 3.3 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaabuaeqale aacaWGRbGaeyicI4Saam4CamaaBaaameaacaWGPbaabeaaaSqab0Ga eyyeIuoakmaalaaabaWaaeWaaeaaceWG3bGbaGaadaWgaaWcbaGaam yAaiaadUgaaeqaaOGaeyOeI0Iabm4DayaaiaWaaSbaaSqaaiaadMga cqGHsislcaaIXaGaaGilaiaaykW7caWGRbaabeaakiaadEhadaWgaa WcbaGaamyAaiaadUgaaeqaaaGccaGLOaGaayzkaaWaaWbaaSqabeaa caaIYaaaaaGcbaGaam4DamaaDaaaleaacaWGPbGaeyOeI0IaaGymai aaiYcacaaMc8Uaam4AaaqaaiaacQcaaaGccaWG3bWaaSbaaSqaaiaa dMgacaWGRbaabeaaaaGccaaMf8UaaGzbVlaaywW7caaMf8UaaGzbVl aacIcacaaIZaGaaiOlaiaaiodacaGGPaaaaa@616C@

c’est-à-dire par des poids non calés au dénominateur. Il est facile de vérifier que les poids calés globaux résultant de la minimisation de (3.3) sous la contrainte (3.2) sont (pour p = 2 , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCaiaai2 dacaaIYaGaaiilaaaa@3722@ voir Hidiroglou et Särndal 1998)

w ˜ p k = w p k * ( g 1 k + + g i k + + g p k ( p 1 ) ) ( 3.4 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabm4Dayaaia WaaSbaaSqaaiaadchacaWGRbaabeaakiaai2dacaWG3bWaa0baaSqa aiaadchacaWGRbaabaGaaiOkaaaakmaabmaabaGaam4zamaaBaaale aacaaIXaGaam4AaaqabaGccqGHRaWkcqWIMaYscqGHRaWkcaWGNbWa aSbaaSqaaiaadMgacaWGRbaabeaakiabgUcaRiablAciljabgUcaRi aadEgadaWgaaWcbaGaamiCaiaadUgaaeqaaOGaeyOeI0YaaeWaaeaa caWGWbGaeyOeI0IaaGymaaGaayjkaiaawMcaaaGaayjkaiaawMcaai aaywW7caaMf8UaaGzbVlaaywW7caaMf8UaaiikaiaaiodacaGGUaGa aGinaiaacMcaaaa@5C34@

g i k = 1 + ( l s i 1 w ˜ i 1, l x i l l s i w ˜ i 1, l w i l x i l ) T i 1 x i k ( 3.5 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4zamaaBa aaleaacaWGPbGaam4AaaqabaGccaaI9aGaaGymaiabgUcaRmaabmaa baWaaabuaeqaleaacaWGSbGaeyicI4Saam4CamaaBaaameaacaWGPb GaeyOeI0IaaGymaaqabaaaleqaniabggHiLdGcceWG3bGbaGaadaWg aaWcbaGaamyAaiabgkHiTiaaigdacaaISaGaaGPaVlaadYgaaeqaaO GaamiEamaaBaaaleaacaWGPbGaamiBaaqabaGccqGHsisldaaeqbqa bSqaaiaadYgacqGHiiIZcaWGZbWaaSbaaWqaaiaadMgaaeqaaaWcbe qdcqGHris5aOGaaGPaVlqadEhagaacamaaBaaaleaacaWGPbGaeyOe I0IaaGymaiaaiYcacaaMc8UaamiBaaqabaGccaWG3bWaaSbaaSqaai aadMgacaWGSbaabeaakiaadIhadaWgaaWcbaGaamyAaiaadYgaaeqa aaGccaGLOaGaayzkaaWaaWbaaSqabeaakiadaITHYaIOaaGaamivam aaDaaaleaacaWGPbaabaGaeyOeI0IaaGymaaaakiaadIhadaWgaaWc baGaamyAaiaadUgaaeqaaOGaaGzbVlaaywW7caaMf8UaaGzbVlaayw W7caGGOaGaaG4maiaac6cacaaI1aGaaiykaaaa@78EA@

pour k s p MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4AaiabgI GiolaadohadaWgaaWcbaGaamiCaaqabaaaaa@3887@ avec T i = l s i w i l * x i l x i l . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamivamaaBa aaleaacaWGPbaabeaakiaai2dadaaeqaqaaiaadEhadaqhaaWcbaGa amyAaiaadYgaaeaacaGGQaaaaOGaamiEamaaBaaaleaacaWGPbGaam iBaaqabaGccaWG4bWaa0baaSqaaiaadMgacaWGSbaabaGccWaGyBOm GikaaiaaygW7aSqaaiaadYgacqGHiiIZcaWGZbWaaSbaaWqaaiaadM gaaeqaaaWcbeqdcqGHris5aOGaaiOlaaaa@4C3F@ Le choix d’une mesure de distance dans la construction des estimateurs calés n’est pas critique, puisque les estimateurs résultants pour une large gamme de mesures de distance sont asymptotiquement équivalents à celui qui utilise la mesure de distance MCG (3.1), Deville et Särndal (1992). Il en est de même de la mesure de distance (3.3). Puisque l’estimateur de Horvitz-Thompson X 1 w 1 * MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiwamaaDa aaleaacaaIXaaabaGccWaGyBOmGikaaiaadEhadaqhaaWcbaGaaGym aaqaaiaacQcaaaaaaa@3B3B@ est sans biais pour t 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiDamaaBa aaleaacaaIXaaabeaaaaa@35DA@ avec un écart-type d’ordre de grandeur N O ( n 1 1 / 2 ) , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOtaiabgw Sixlaad+eadaqadaqaaiaad6gadaqhaaWcbaGaaGymaaqaaiabgkHi TmaalyaabaGaaGymaaqaaiaaikdaaaaaaaGccaGLOaGaayzkaaGaai ilaaaa@3E83@ alors g 1 k = 1 + O ( n 1 1 / 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4zamaaBa aaleaacaaIXaGaam4AaaqabaGccaaI9aGaaGymaiabgUcaRiaad+ea daqadaqaaiaad6gadaqhaaWcbaGaaGymaaqaaiabgkHiTmaalyaaba GaaGymaaqaaiaaikdaaaaaaaGccaGLOaGaayzkaaaaaa@3FE7@ pour tout k s 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4AaiabgI GiolaadohadaWgaaWcbaGaaGymaaqabaaaaa@384D@ et donc w ˜ 1 k = w 1 k * ( 1 + O ( n 1 1 / 2 ) ) . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabm4Dayaaia WaaSbaaSqaaiaaigdacaWGRbaabeaakiaai2dacaWG3bWaa0baaSqa aiaaigdacaWGRbaabaGaaiOkaaaakmaabmaabaGaaGymaiabgUcaRi aad+eadaqadaqaaiaad6gadaqhaaWcbaGaaGymaaqaaiabgkHiTmaa lyaabaGaaGymaaqaaiaaikdaaaaaaaGccaGLOaGaayzkaaaacaGLOa GaayzkaaGaaiOlaaaa@45CD@ Par induction, g i k = 1 + O ( n i 1 / 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4zamaaBa aaleaacaWGPbGaam4AaaqabaGccaaI9aGaaGymaiabgUcaRiaad+ea daqadaqaaiaad6gadaqhaaWcbaGaamyAaaqaaiabgkHiTmaalyaaba GaaGymaaqaaiaaikdaaaaaaaGccaGLOaGaayzkaaaaaa@404D@ pour tout i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAaaaa@34E8@ et découlant de (3.4), w ˜ p k / w p k * 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSGbaeaace WG3bGbaGaadaWgaaWcbaGaamiCaiaadUgaaeqaaaGcbaGaam4Damaa DaaaleaacaWGWbGaam4AaaqaaiaacQcaaaaaaOGaeyOKH4QaaGymaa aa@3DA4@ en probabilité avec n p . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBamaaBa aaleaacaWGWbaabeaakiaac6caaaa@36CA@ Suggérant de nouvelles techniques en vue d’améliorer l’estimation, Farrell et Singh (2002) ont proposé d’autres types de fonction de distance du khi carré pénalisée.

3.1 Estimation

L’analyse qui suit est motivée par la nature récursive de w ˜ i k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabm4Dayaaia WaaSbaaSqaaiaadMgacaWGRbaabeaaaaa@370F@ dans (3.4), où les poids calés des phases antérieures 1, , i 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiaaiY cacqWIMaYscaaISaGaamyAaiabgkHiTiaaigdaaaa@39D9@ sont emboîtés dans chaque facteur g i k , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4zamaaBa aaleaacaWGPbGaam4AaaqabaGccaGGSaaaaa@37AA@ ce qui requiert le calcul séquentiel des poids calés; autrement dit, il faut calculer tous les poids calés des phases antérieures pour obtenir ceux des phases ultérieures. Soient B ^ i j + = ( k s i w i k * x i k x i k ) 1 k s j w j k * x i k x j k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOqayaaja Waa0baaSqaaiaadMgacaWGQbaabaGaey4kaScaaOGaaGypamaabmqa baWaaabeaeaacaWG3bWaa0baaSqaaiaadMgacaWGRbaabaGaaiOkaa aakiaadIhadaWgaaWcbaGaamyAaiaadUgaaeqaaOGaamiEamaaDaaa leaacaWGPbGaam4AaaqaaOGamai2gkdiIcaaaSqaaiaadUgacqGHii IZcaWGZbWaaSbaaWqaaiaadMgaaeqaaaWcbeqdcqGHris5aaGccaGL OaGaayzkaaWaaWbaaSqabeaacqGHsislcaaIXaaaaOWaaabeaeaaca WG3bWaa0baaSqaaiaadQgacaWGRbaabaGaaiOkaaaakiaadIhadaWg aaWcbaGaamyAaiaadUgaaeqaaOGaamiEamaaDaaaleaacaWGQbGaam 4AaaqaaOGamai2gkdiIcaaaSqaaiaadUgacqGHiiIZcaWGZbWaaSba aWqaaiaadQgaaeqaaaWcbeqdcqGHris5aaaa@6271@ et B ^ i j = ( k s i w i k * x i k x i k ) 1 k s j 1 w j 1, k * x i k x j k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOqayaaja Waa0baaSqaaiaadMgacaWGQbaabaGaeyOeI0caaOGaaGypamaabmqa baWaaabeaeqaleaacaWGRbGaeyicI4Saam4CamaaBaaameaacaWGPb aabeaaaSqab0GaeyyeIuoakiaadEhadaqhaaWcbaGaamyAaiaadUga aeaacaGGQaaaaOGaamiEamaaBaaaleaacaWGPbGaam4AaaqabaGcca WG4bWaa0baaSqaaiaadMgacaWGRbaabaGccWaGyBOmGikaaaGaayjk aiaawMcaamaaCaaaleqabaGaeyOeI0IaaGymaaaakmaaqababeWcba Gaam4AaiabgIGiolaadohadaWgaaadbaGaamOAaiabgkHiTiaaigda aeqaaaWcbeqdcqGHris5aOGaam4DamaaDaaaleaacaWGQbGaeyOeI0 IaaGymaiaaiYcacaaMc8Uaam4AaaqaaiaacQcaaaGccaWG4bWaaSba aSqaaiaadMgacaWGRbaabeaakiaadIhadaqhaaWcbaGaamOAaiaadU gaaeaakiadaITHYaIOaaaaaa@681A@ les estimateurs de B i j = ( k U x i k x i k ) 1 k U x i k x j k , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOqamaaBa aaleaacaWGPbGaamOAaaqabaGccaaI9aWaaeWabeaadaaeqaqaaiaa dIhadaWgaaWcbaGaamyAaiaadUgaaeqaaOGaamiEamaaDaaaleaaca WGPbGaam4AaaqaaOGamai2gkdiIcaaaSqaaiaadUgacqGHiiIZcaWG vbaabeqdcqGHris5aaGccaGLOaGaayzkaaWaaWbaaSqabeaacqGHsi slcaaIXaaaaOWaaabeaeaacaWG4bWaaSbaaSqaaiaadMgacaWGRbaa beaakiaadIhadaqhaaWcbaGaamOAaiaadUgaaeaakiadaITHYaIOaa GaaGzaVlaacYcaaSqaaiaadUgacqGHiiIZcaWGvbaabeqdcqGHris5 aaaa@59B1@ le coefficient de régression de x j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCiEamaaBa aaleaacaWGQbaabeaaaaa@3616@ sur x i . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCiEamaaBa aaleaacaWGPbaabeaakiaac6caaaa@36D1@ La différence entre les deux estimateurs tient au fait que, tandis que B ^ i j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOqayaaja Waa0baaSqaaiaadMgacaWGQbaabaGaeyOeI0caaaaa@37C8@ utilise l’ensemble complet d’unités connues pour x j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCiEamaaBa aaleaacaWGQbaabeaaaaa@3616@ qui est obtenu dans s j 1 , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4CamaaBa aaleaacaWGQbGaeyOeI0IaaGymaaqabaGccaGGSaaaaa@386F@ B ^ i j + MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOqayaaja Waa0baaSqaaiaadMgacaWGQbaabaGaey4kaScaaaaa@37BD@ utilise uniquement le sous-ensemble s j s j 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4CamaaBa aaleaacaWGQbaabeaakiabgAOinlaadohadaWgaaWcbaGaamOAaiab gkHiTiaaigdaaeqaaaaa@3BD3@ et, donc, plus de variables que B ^ i j . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOqayaaja Waa0baaSqaaiaadMgacaWGQbaabaGaeyOeI0caaOGaaiOlaaaa@3884@ Soit Z ^ i j = B ^ i j + B ^ i j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOwayaaja WaaSbaaSqaaiaadMgacaWGQbaabeaakiaai2daceWGcbGbaKaadaqh aaWcbaGaamyAaiaadQgaaeaacqGHRaWkaaGccqGHsislceWGcbGbaK aadaqhaaWcbaGaamyAaiaadQgaaeaacqGHsislaaaaaa@404B@ la différence entre les deux coefficients estimés qui converge vers zéro. Notons aussi Z ^ i 1 i 2 i k = j = 2 k Z ^ i j 1 i j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOwayaaja WaaSbaaSqaaiaadMgadaWgaaadbaGaaGymaaqabaWccaWGPbWaaSba aWqaaiaaikdaaeqaaSGaeSOjGSKaamyAamaaBaaameaacaWGRbaabe aaaSqabaGccaaI9aWaaebmaeaaceWGAbGbaKaadaWgaaWcbaGaamyA amaaBaaameaacaWGQbGaeyOeI0IaaGymaaqabaWccaWGPbWaaSbaaW qaaiaadQgaaeqaaaWcbeaaaeaacaWGQbGaaGypaiaaikdaaeaacaWG Rbaaniabg+Givdaaaa@4916@ pour k 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4Aaiabgw MiZkaaikdaaaa@376C@ et Z ^ i 1 = 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOwayaaja WaaSbaaSqaaiaadMgadaWgaaadbaGaaGymaaqabaaaleqaaOGaaGyp aiaaigdaaaa@3881@ pour k = 1. MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4Aaiaai2 dacaaIXaGaaiOlaaaa@371E@ Soit t ^ i = k s i 1 w i 1 k * x i k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmiDayaaja Waa0baaSqaaiaadMgaaeaacqGHsislaaGccaaI9aWaaabeaeaacaWG 3bWaa0baaSqaaiaadMgacqGHsislcaaIXaGaam4AaaqaaiaacQcaaa GccaWG4bWaaSbaaSqaaiaadMgacaWGRbaabeaaaeaacaWGRbGaeyic I4Saam4CamaaBaaameaacaWGPbGaeyOeI0IaaGymaaqabaaaleqani abggHiLdaaaa@485C@ et t ^ i + = k s i w i k * x i k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmiDayaaja Waa0baaSqaaiaadMgaaeaacqGHRaWkaaGccaaI9aWaaabeaeaacaWG 3bWaa0baaSqaaiaadMgacaWGRbaabaGaaiOkaaaakiaadIhadaWgaa WcbaGaamyAaiaadUgaaeqaaaqaaiaadUgacqGHiiIZcaWGZbWaaSba aWqaaiaadMgaaeqaaaWcbeqdcqGHris5aaaa@4501@ les deux estimateurs de Horvitz-Thompson pour t i , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiDamaaBa aaleaacaWGPbaabeaakiaacYcaaaa@36C7@ fondés sur les unités obtenues dans les échantillons s i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4CamaaBa aaleaacaWGPbaabeaaaaa@360C@ et s i 1 , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4CamaaBa aaleaacaWGPbGaeyOeI0IaaGymaaqabaGccaGGSaaaaa@386E@ respectivement. Notons que tous les estimateurs définis dans le présent paragraphe utilisent les poids de sondage globaux w * MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4DamaaCa aaleqabaGaaiOkaaaaaaa@35D1@ et non les poids calés. Dans le lemme qui suit, nous donnons une représentation de w ˜ p , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabm4Dayaaia WaaSbaaSqaaiaadchaaeqaaOGaaiilaaaa@36E0@ le vecteur de poids calés après p MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCaaaa@34EF@ phases de calage, qui dépend uniquement des poids de sondage connus au préalable { w i * } i = 1 p . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiWaaeaaca WG3bWaa0baaSqaaiaadMgaaeaacaGGQaaaaaGccaGL7bGaayzFaaWa a0baaSqaaiaadMgacaaI9aGaaGymaaqaaiaadchaaaGccaaMb8Uaai Olaaaa@3ED2@

Lemme 3.1 Considérons un plan d’échantillonnage à plusieurs phases avec un scénario de calage qui produit des facteurs g additifs comme il est défini dans (3.3). Une représentation des poids calés à la phase  p MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCaaaa@34EF@ fondée entièrement sur les poids de sondage est

w ˜ p = D p *′ 1 n p + i 1 = 1 p A i 1 i 1 < i 2 p A i 1 i 2 + + ( 1 ) k + 1 i 1 < < i k p A i 1 i 2 i k + + ( 1 ) p + 1 A i 1 i 2 i p ( 3.6 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqbaeaabiGaaa qaaiqadEhagaacamaaBaaaleaacaWGWbaabeaaaOqaaiaai2dacaWG ebWaa0baaSqaaiaadchaaeaacaGGQaGccWaGyBOmGikaaiaaigdada WgaaWcbaGaamOBamaaBaaameaacaWGWbaabeaaaSqabaGccqGHRaWk daaeWbqaaiaadgeadaWgaaWcbaGaamyAamaaBaaameaacaaIXaaabe aaaSqabaaabaGaamyAamaaBaaameaacaaIXaaabeaaliaai2dacaaI XaaabaGaamiCaaqdcqGHris5aOGaeyOeI0YaaabCaeaacaWGbbWaaS baaSqaaiaadMgadaWgaaadbaGaaGymaaqabaWccaWGPbWaaSbaaWqa aiaaikdaaeqaaaWcbeaaaeaacaWGPbWaaSbaaWqaaiaaigdaaeqaaS GaaGipaiaadMgadaWgaaadbaGaaGOmaaqabaaaleaacaWGWbaaniab ggHiLdaakeaaaeaacaaMc8UaaGPaVlabgUcaRiablAciljabgUcaRm aabmaabaGaeyOeI0IaaGymaaGaayjkaiaawMcaamaaCaaaleqabaGa am4AaiabgUcaRiaaigdaaaGcdaaeWbqaaiaadgeadaWgaaWcbaGaam yAamaaBaaameaacaaIXaaabeaaliaadMgadaWgaaadbaGaaGOmaaqa baWccqWIMaYscaWGPbWaaSbaaWqaaiaadUgaaeqaaaWcbeaaaeaaca WGPbWaaSbaaWqaaiaaigdaaeqaaSGaaGipaiablAciljaaiYdacaWG PbWaaSbaaWqaaiaadUgaaeqaaaWcbaGaamiCaaqdcqGHris5aOGaey 4kaSIaeSOjGSKaey4kaSYaaeWaaeaacqGHsislcaaIXaaacaGLOaGa ayzkaaWaaWbaaSqabeaacaWGWbGaey4kaSIaaGymaaaakiaadgeada WgaaWcbaGaamyAamaaBaaameaacaaIXaaabeaaliaadMgadaWgaaad baGaaGOmaaqabaWccqWIMaYscaWGPbWaaSbaaWqaaiaadchaaeqaaa WcbeaakiaaywW7caaMf8UaaGzbVlaaywW7caaMf8Uaaiikaiaaioda caGGUaGaaGOnaiaacMcaaaaaaa@912B@

A i 1 i 2 i k = ( t ^ i 1 t ^ i 1 + ) Z ^ i 1 i 2 i k ( X i k D i k * X i k ) 1 X i k D p * . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqamaaDa aaleaacaWGPbWaaSbaaWqaaiaaigdaaeqaaSGaamyAamaaBaaameaa caaIYaaabeaaliablAciljaadMgadaWgaaadbaGaam4Aaaqabaaale aakiadaITHYaIOaaGaaGypamaabmaabaGabmiDayaajaWaa0baaSqa aiaadMgadaWgaaadbaGaaGymaaqabaaaleaacqGHsislaaGccqGHsi slceWG0bGbaKaadaqhaaWcbaGaamyAamaaBaaameaacaaIXaaabeaa aSqaaiabgUcaRaaaaOGaayjkaiaawMcaamaaCaaaleqabaGccWaGyB OmGikaaiqadQfagaqcamaaBaaaleaacaWGPbWaaSbaaWqaaiaaigda aeqaaSGaamyAamaaBaaameaacaaIYaaabeaaliablAciljaadMgada WgaaadbaGaam4AaaqabaaaleqaaOWaaeWaaeaacaWGybWaa0baaSqa aiaadMgadaWgaaadbaGaam4AaaqabaaaleaakiadaITHYaIOaaGaam iramaaDaaaleaacaWGPbWaaSbaaWqaaiaadUgaaeqaaaWcbaGaaiOk aaaakiaadIfadaWgaaWcbaGaamyAamaaBaaameaacaWGRbaabeaaaS qabaaakiaawIcacaGLPaaadaahaaWcbeqaaiabgkHiTiaaigdaaaGc caWGybWaa0baaSqaaiaadMgadaWgaaadbaGaam4Aaaqabaaaleaaki adaITHYaIOaaGaamiramaaDaaaleaacaWGWbaabaGaaiOkaaaakiaa ygW7caaIUaaaaa@70A4@

Preuve. Voir l’annexe A.

Notons la forme « inclusion-exclusion » de w ˜ p MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabm4Dayaaia WaaSbaaSqaaiaadchaaeqaaaaa@3626@ dans le lemme 3.1. La k e MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4AamaaCa aaleqabaGccWaGyBOmGi6ccaqGLbaaaaaa@38F4@ sommation comprend ( p k ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpK0df9Wqpm0db9Wq pepeuf0xe9q8qiYRWFGCk9vi=dbvc9s8vr0=qr0db9q8qi0Je9Fve9 Fve9FXqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaabmaabaqbae aabiqaaaqaaKqzaeGaamiCaaGcbaqcLbqacaWGRbaaaaGccaGLOaGa ayzkaaaaaa@3DC7@ opérandes A i 1 i 2 i k , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqamaaBa aaleaacaWGPbWaaSbaaWqaaiaaigdaaeqaaSGaamyAamaaBaaameaa caaIYaaabeaaliablAciljaadMgadaWgaaadbaGaam4Aaaqabaaale qaaOGaaGzaVlaacYcaaaa@3E2B@ pour lesquels chaque Z ^ i 1 i 2 i k = j = 2 k ( B ^ i j 1 i j + B ^ i j 1 i j ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOwayaaja WaaSbaaSqaaiaadMgadaWgaaadbaGaaGymaaqabaWccaWGPbWaaSba aWqaaiaaikdaaeqaaSGaeSOjGSKaamyAamaaBaaameaacaWGRbaabe aaaSqabaGccaaI9aWaaebmaeaadaqadaqaaiqadkeagaqcamaaDaaa leaacaWGPbWaaSbaaWqaaiaadQgacqGHsislcaaIXaaabeaaliaadM gadaWgaaadbaGaamOAaaqabaaaleaacqGHRaWkaaGccqGHsislceWG cbGbaKaadaqhaaWcbaGaamyAamaaBaaameaacaWGQbGaeyOeI0IaaG ymaaqabaWccaWGPbWaaSbaaWqaaiaadQgaaeqaaaWcbaGaeyOeI0ca aaGccaGLOaGaayzkaaaaleaacaWGQbGaaGypaiaaikdaaeaacaWGRb aaniabg+Givdaaaa@5439@ contient 2 k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOmamaaCa aaleqabaGaam4Aaaaaaaa@35D3@ opérandes. Soit, un total de ( p k ) 2 k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpK0df9Wqpm0db9Wq pepeuf0xe9q8qiYRWFGCk9vi=dbvc9s8vr0db9Ff0dbbG8Fq0Jfr=x fr=xfbpdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaafa qaaeGabaaabaqcLbqacaWGWbaakeaajugabiaadUgaaaaakiaawIca caGLPaaacaaIYaWaaWbaaSqabeaajugZaiaadUgaaaaaaa@40CC@ opérandes. Le nombre global de termes dans (3.6) est par conséquent 3 p MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaG4mamaaCa aaleqabaGaamiCaaaaaaa@35D9@ comme il est montré dans la preuve du lemme. Notons aussi que les termes A i 1 i 2 i k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqamaaBa aaleaacaWGPbWaaSbaaWqaaiaaigdaaeqaaSGaamyAamaaBaaameaa caaIYaaabeaaliablAciljaadMgadaWgaaadbaGaam4Aaaqabaaale qaaaaa@3BE7@ comprennent le produit des composantes t ^ i 1 t ^ i 1 + MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmiDayaaja Waa0baaSqaaiaadMgadaWgaaadbaGaaGymaaqabaaaleaacqGHsisl aaGccqGHsislceWG0bGbaKaadaqhaaWcbaGaamyAamaaBaaameaaca aIXaaabeaaaSqaaiabgUcaRaaaaaa@3CEE@ et Z ^ i 1 i 2 i k , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOwayaaja WaaSbaaSqaaiaadMgadaWgaaadbaGaaGymaaqabaWccaWGPbWaaSba aWqaaiaaikdaaeqaaSGaeSOjGSKaamyAamaaBaaameaacaWGRbaabe aaaSqabaGccaaMb8Uaaiilaaaa@3E54@ ayant toutes deux une espérance nulle, de sorte que le poids calé w ˜ p MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabm4Dayaaia WaaSbaaSqaaiaadchaaeqaaaaa@3626@ est égal à D p *′ 1 n p , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiramaaDa aaleaacaWGWbaabaGaaiOkaOGamai2gkdiIcaacaaIXaWaaSbaaSqa aiaad6gadaWgaaadbaGaamiCaaqabaaaleqaaOGaaGzaVlaacYcaaa a@3EC8@ le poids de sondage global, plus les termes de correction d’ordres de grandeur plus faibles, et maintient la caractéristique bien connue des poids calés. Jusqu’à présent, nous nous sommes limités dans notre discussion à une représentation du vecteur des poids dans un processus de calage à plusieurs phases qui fait intervenir uniquement des paramètres du plan de sondage et n’inclut pas les facteurs g . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4zaiaac6 caaaa@35B8@ Or, partant de cette représentation de w ˜ p , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabm4Dayaaia WaaSbaaSqaaiaadchaaeqaaOGaaiilaaaa@36E0@ il est possible de déduire un estimateur novateur pour la variance des estimateurs calés en plusieurs phases. Soit y MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEaaaa@34F8@ une variable d’intérêt pour laquelle nous voulons estimer le total de population Y . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamywaiaac6 caaaa@358A@ Soit β ^ j = ( k s j w j k * x j k x j k ) 1 k s p w p k * x j k y k , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqOSdiMbaK aadaWgaaWcbaGaamOAaaqabaGccaaI9aWaaeWaaeaadaaeqaqaaiaa dEhadaqhaaWcbaGaamOAaiaadUgaaeaacaGGQaaaaOGaamiEamaaBa aaleaacaWGQbGaam4AaaqabaGccaWG4bWaa0baaSqaaiaadQgacaWG RbaabaGccWaGyBOmGikaaaWcbaGaam4AaiabgIGiolaadohadaWgaa adbaGaamOAaaqabaaaleqaniabggHiLdaakiaawIcacaGLPaaadaah aaWcbeqaaiabgkHiTiaaigdaaaGcdaaeqaqabSqaaiaadUgacqGHii IZcaWGZbWaaSbaaWqaaiaadchaaeqaaaWcbeqdcqGHris5aOGaam4D amaaDaaaleaacaWGWbGaam4AaaqaaiaacQcaaaGccaWG4bWaaSbaaS qaaiaadQgacaWGRbaabeaakiaadMhadaWgaaWcbaGaam4AaaqabaGc caaMb8Uaaiilaaaa@6001@ le coefficient de régression de y MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEaaaa@34F8@ sur x j , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCiEamaaBa aaleaacaWGQbaabeaakiaacYcaaaa@36D0@ et Y ^ HT p = 1 n p D p * y , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmywayaaja WaaSbaaSqaaiaabIeacaqGubWaaSbaaWqaaiaadchaaeqaaaWcbeaa kiaai2dacaaIXaWaa0baaSqaaiaad6gadaWgaaadbaGaamiCaaqaba aaleaakiadaITHYaIOaaGaamiramaaDaaaleaacaWGWbaabaGaaiOk aaaakiaadMhacaGGSaaaaa@42F7@ l’estimateur de Horvitz-Thompson non calé, calculé sur les éléments compris dans s p . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4CamaaBa aaleaacaWGWbaabeaakiaac6caaaa@36CF@ Le réarrangement des termes dans (3.6) produit une représentation plus classique de l’estimateur calé en plusieurs phases w ˜ p y MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabm4Dayaaia Waa0baaSqaaiaadchaaeaakiadaITHYaIOaaGaamyEaaaa@3A0F@ sous forme d’un estimateur par la régression multivariée

w ˜ p y = Y ^ HT p + i 1 = 1 p ( t ^ i 1 t ^ i 1 + ) γ ^ i 1 ( 3.7 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabm4Dayaaia Waa0baaSqaaiaadchaaeaakiadaITHYaIOaaGaamyEaiaai2daceWG zbGbaKaadaWgaaWcbaGaaeisaiaabsfadaWgaaadbaGaamiCaaqaba aaleqaaOGaey4kaSYaaabCaeaadaqadaqaaiqadshagaqcamaaDaaa leaacaWGPbWaaSbaaWqaaiaaigdaaeqaaaWcbaGaeyOeI0caaOGaey OeI0IabmiDayaajaWaa0baaSqaaiaadMgadaWgaaadbaGaaGymaaqa baaaleaacqGHRaWkaaaakiaawIcacaGLPaaadaahaaWcbeqaaOGama i2gkdiIcaacuaHZoWzgaqcamaaBaaaleaacaWGPbWaaSbaaWqaaiaa igdaaeqaaaWcbeaaaeaacaWGPbWaaSbaaWqaaiaaigdaaeqaaSGaaG ypaiaaigdaaeaacaWGWbaaniabggHiLdGccaaMf8UaaGzbVlaaywW7 caaMf8UaaGzbVlaacIcacaaIZaGaaiOlaiaaiEdacaGGPaaaaa@62F4@

γ ^ i 1 = β ^ i 1 i 1 < i 2 p Z ^ i 1 i 2 β ^ i 2 + + ( 1 ) k + 1 i 1 < < i k p Z ^ i 1 i 2 i k β ^ i k + + ( 1 ) p ( i 1 1 ) + 1 Z ^ i 1 p β ^ p . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipu0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqbaeaabiGaaa qaaiqbeo7aNzaajaWaaSbaaSqaaiaadMgadaWgaaadbaGaaGymaaqa baaaleqaaaGcbaGaaGypaiqbek7aIzaajaWaaSbaaSqaaiaadMgada WgaaadbaGaaGymaaqabaaaleqaaOGaeyOeI0YaaabCaeaaceWGAbGb aKaadaWgaaWcbaGaamyAamaaBaaameaacaaIXaaabeaaliaadMgada WgaaadbaGaaGOmaaqabaaaleqaaOGafqOSdiMbaKaadaWgaaWcbaGa amyAamaaBaaameaacaaIYaaabeaaaSqabaGccqGHRaWkaSqaaiaadM gadaWgaaadbaGaaGymaaqabaWccaaI8aGaamyAamaaBaaameaacaaI YaaabeaaaSqaaiaadchaa0GaeyyeIuoaaOqaaaqaaiaaykW7caaMc8 UaaGPaVlaaykW7cqWIMaYscqGHRaWkdaqadaqaaiabgkHiTiaaigda aiaawIcacaGLPaaadaahaaWcbeqaaiaadUgacqGHRaWkcaaIXaaaaO WaaabCaeaaceWGAbGbaKaadaWgaaWcbaGaamyAamaaBaaameaacaaI XaaabeaaliaadMgadaWgaaadbaGaaGOmaaqabaWccqWIMaYscaWGPb WaaSbaaWqaaiaadUgaaeqaaaWcbeaakiqbek7aIzaajaWaaSbaaSqa aiaadMgadaWgaaadbaGaam4AaaqabaaaleqaaaqaaiaadMgadaWgaa adbaGaaGymaaqabaWccaaI8aGaeSOjGSKaaGipaiaadMgadaWgaaad baGaam4AaaqabaaaleaacaWGWbaaniabggHiLdGccqGHRaWkcqWIMa YscqGHRaWkdaqadaqaaiabgkHiTiaaigdaaiaawIcacaGLPaaadaah aaWcbeqaaiaadchacqGHsisldaqadaqaaiaadMgadaWgaaadbaGaaG ymaaqabaWccqGHsislcaaIXaaacaGLOaGaayzkaaGaey4kaSIaaGym aaaakiqadQfagaqcamaaBaaaleaacaWGPbWaaSbaaWqaaiaaigdaae qaaSGaeSOjGSKaamiCaaqabaGccuaHYoGygaqcamaaBaaaleaacaWG Wbaabeaakiaai6caaaaaaa@8A0D@

L’établissement d’un estimateur convergent de la variance des estimateurs calés en plusieurs phases est maintenant simple en ce sens qu’il suit à peu près les étapes utilisées dans le calcul de la variance sous un scénario de calage multivarié à une phase.

Théorème 3.1 Soit e ^ r k = x r k γ ^ r x r + 1, k γ ^ r + 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmyzayaaja WaaSbaaSqaaiaadkhacaWGRbaabeaakiaai2dacaWG4bWaa0baaSqa aiaadkhacaWGRbaabaGccWaGyBOmGikaaiqbeo7aNzaajaWaaSbaaS qaaiaadkhaaeqaaOGaeyOeI0IaamiEamaaDaaaleaacaWGYbGaey4k aSIaaGymaiaaiYcacaaMc8Uaam4AaaqaaOGamai2gkdiIcaacuaHZo WzgaqcamaaBaaaleaacaWGYbGaey4kaSIaaGymaaqabaaaaa@4FF4@  pour r < p MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCaiaaiY dacaWGWbaaaa@36AC@  et e ^ p k = x p k γ ^ p y k . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmyzayaaja WaaSbaaSqaaiaadchacaWGRbaabeaakiaai2dacaWG4bWaa0baaSqa aiaadchacaWGRbaabaGccWaGyBOmGikaaiqbeo7aNzaajaWaaSbaaS qaaiaadchaaeqaaOGaeyOeI0IaamyEamaaBaaaleaacaWGRbaabeaa kiaac6caaaa@4474@  Un estimateur convergent de la variance de w ˜ p  ′ y MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabm4Dayaaia Waa0baaSqaaiaadchaaeaakmaaCaaameqabaqcLbwacWaGyBOmGika aaaakiaadMhaaaa@3B16@  est

1 r 1 , r 2 p k s r m , l s r M w r M l * w r m l * ( w r m k * w r m l * w r m k l * ) e ^ r m k e ^ r M l ( 3.8 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaabuaeaaca aMe8+aaabuaeaadaWcaaqaaiaadEhadaqhaaWcbaGaamOCamaaBaaa meaacaWGnbaabeaaliaadYgaaeaacaGGQaaaaaGcbaGaam4DamaaDa aaleaacaWGYbWaaSbaaWqaaiaad2gaaeqaaSGaamiBaaqaaiaacQca aaaaaaqaaiaadUgacqGHiiIZcaWGZbWaaSbaaWqaaiaadkhadaWgaa qaaiaad2gaaeqaaaqabaWccaaMb8UaaGilaiaaysW7caWGSbGaeyic I4Saam4CamaaBaaameaacaWGYbWaaSbaaeaacaWGnbaabeaaaeqaaa WcbeqdcqGHris5aaWcbaGaaGymaiabgsMiJkaadkhadaWgaaadbaGa aGymaaqabaWccaaMb8UaaGilaiaaysW7caWGYbWaaSbaaWqaaiaaik daaeqaaSGaeyizImQaamiCaaqab0GaeyyeIuoakmaabmaabaGaam4D amaaDaaaleaacaWGYbWaaSbaaWqaaiaad2gaaeqaaSGaam4Aaaqaai aacQcaaaGccaWG3bWaa0baaSqaaiaadkhadaWgaaadbaGaamyBaaqa baWccaWGSbaabaGaaiOkaaaakiabgkHiTiaadEhadaqhaaWcbaGaam OCamaaBaaameaacaWGTbaabeaaliaadUgacaWGSbaabaGaaiOkaaaa aOGaayjkaiaawMcaaiqadwgagaqcamaaBaaaleaacaWGYbWaaSbaaW qaaiaad2gaaeqaaSGaam4AaaqabaGcceWGLbGbaKaadaWgaaWcbaGa amOCamaaBaaameaacaWGnbaabeaaliaadYgaaeqaaOGaaGzbVlaayw W7caaMf8UaaGzbVlaaywW7caGGOaGaaG4maiaac6cacaaI4aGaaiyk aaaa@854E@

r m = min ( r 1 , r 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCamaaBa aaleaacaWGTbaabeaakiaai2daciGGTbGaaiyAaiaac6gadaqadaqa aiaadkhadaWgaaWcbaGaaGymaaqabaGccaaMb8UaaGilaiaaykW7ca WGYbWaaSbaaSqaaiaaikdaaeqaaaGccaGLOaGaayzkaaaaaa@42D7@  et r M = max ( r 1 , r 2 ) . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCamaaBa aaleaacaWGnbaabeaakiaai2daciGGTbGaaiyyaiaacIhadaqadaqa aiaadkhadaWgaaWcbaGaaGymaaqabaGccaaMb8UaaGilaiaaykW7ca WGYbWaaSbaaSqaaiaaikdaaeqaaaGccaGLOaGaayzkaaGaaiOlaaaa @436B@

La preuve comprend l’évaluation des ordres de grandeur les plus élevés et l’estimation de leur variance. Une attention particulière est accordée à l’évaluation de la probabilité conjointe des événements { k s i , l s j } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiWaaeaaca WGRbGaeyicI4Saam4CamaaBaaaleaacaWGPbaabeaakiaaygW7caaI SaGaaGPaVlaadYgacqGHiiIZcaWGZbWaaSbaaSqaaiaadQgaaeqaaa GccaGL7bGaayzFaaaaaa@4318@ et à l’estimation de la covariance entre les unités provenant de différentes phases d’échantillonnage.

Preuve. À la première étape, nous allons voir que le remplacement des estimateurs des coefficients γ ^ i ; i = 1 p MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafq4SdCMbaK aadaWgaaWcbaGaamyAaaqabaGccaGG7aGaaGjbVlaadMgacaaI9aGa aGymaiablAciljaadchaaaa@3DA8@ par leurs valeurs réelles γ i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4SdC2aaS baaSqaaiaadMgaaeqaaaaa@36BB@ affecte l’estimation de la variance d’un facteur N 2 o ( n p 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOtamaaCa aaleqabaGaaGOmaaaakiaad+gadaqadeqaaiaad6gadaqhaaWcbaGa amiCaaqaaiabgkHiTiaaigdaaaaakiaawIcacaGLPaaaaaa@3C05@ et, donc, n’affecte pas la convergence de l’estimateur substitué. À cette fin, notons que B ^ i j + , B ^ i j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOqayaaja Waa0baaSqaaiaadMgacaWGQbaabaGaey4kaScaaOGaaGilaiaaysW7 ceWGcbGbaKaadaqhaaWcbaGaamyAaiaadQgaaeaacqGHsislaaaaaa@3DD8@ sont tous deux convergents vers B i j . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOqamaaBa aaleaacaWGPbGaamOAaaqabaGccaGGUaaaaa@3786@ Écrivons B ^ i j + = B i j + ( B ^ i j + B i j ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOqayaaja Waa0baaSqaaiaadMgacaWGQbaabaGaey4kaScaaOGaaGypaiaadkea daWgaaWcbaGaamyAaiaadQgaaeqaaOGaey4kaSYaaeWabeaaceWGcb GbaKaadaqhaaWcbaGaamyAaiaadQgaaeaacqGHRaWkaaGccqGHsisl caWGcbWaaSbaaSqaaiaadMgacaWGQbaabeaaaOGaayjkaiaawMcaaa aa@4568@ de sorte que B ^ i j + = B i j + O p ( n j 1 / 2 ) . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOqayaaja Waa0baaSqaaiaadMgacaWGQbaabaGaey4kaScaaOGaaGypaiaadkea daWgaaWcbaGaamyAaiaadQgaaeqaaOGaey4kaSIaam4tamaaBaaale aacaWGWbaabeaakmaabmqabaGaamOBamaaDaaaleaacaWGQbaabaGa eyOeI0YaaSGbaeaacaaIXaaabaGaaGOmaaaaaaaakiaawIcacaGLPa aacaGGUaaaaa@4518@ Rappelons que Z ^ i j = B ^ i j + B ^ i j , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOwayaaja WaaSbaaSqaaiaadMgacaWGQbaabeaakiaai2daceWGcbGbaKaadaqh aaWcbaGaamyAaiaadQgaaeaacqGHRaWkaaGccqGHsislceWGcbGbaK aadaqhaaWcbaGaamyAaiaadQgaaeaacqGHsislaaGccaGGSaaaaa@4105@ B ^ i j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOqayaaja Waa0baaSqaaiaadMgacaWGQbaabaGaeyOeI0caaaaa@37C8@ est basé sur s j 1 , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4CamaaBa aaleaacaWGQbGaeyOeI0IaaGymaaqabaGccaGGSaaaaa@386F@ tandis que B ^ i j + MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOqayaaja Waa0baaSqaaiaadMgacaWGQbaabaGaey4kaScaaaaa@37BD@ est basé sur son sous-échantillon s j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4CamaaBa aaleaacaWGQbaabeaaaaa@360D@ et, donc, Z ^ i j = O p ( n j 1 / 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOwayaaja WaaSbaaSqaaiaadMgacaWGQbaabeaakiaai2dacaWGpbWaaSbaaSqa aiaadchaaeqaaOWaaeWabeaacaWGUbWaa0baaSqaaiaadQgaaeaacq GHsisldaWcgaqaaiaaigdaaeaacaaIYaaaaaaaaOGaayjkaiaawMca aaaa@3FDF@ et, par conséquent, Z ^ i 1 i 2 i k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOwayaaja WaaSbaaSqaaiaadMgadaWgaaadbaGaaGymaaqabaWccaWGPbWaaSba aWqaaiaaikdaaeqaaSGaeSOjGSKaamyAamaaBaaameaacaWGRbaabe aaaSqabaaaaa@3C10@ est borné par O p ( n i k 1 / 2 ) . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4tamaaBa aaleaacaWGWbaabeaakmaabmqabaGaamOBamaaDaaaleaacaWGPbWa aSbaaWqaaiaadUgaaeqaaaWcbaGaeyOeI0YaaSGbaeaacaaIXaaaba GaaGOmaaaaaaaakiaawIcacaGLPaaacaGGUaaaaa@3DEF@ De même, β ^ j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqOSdiMbaK aadaWgaaWcbaGaamOAaaqabaaaaa@36C6@ est β j + O p ( n p 1 / 2 ) , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOSdi2aaS baaSqaaiaadQgaaeqaaOGaey4kaSIaam4tamaaBaaaleaacaWGWbaa beaakmaabmqabaGaamOBamaaDaaaleaacaWGWbaabaGaeyOeI0YaaS GbaeaacaaIXaaabaGaaGOmaaaaaaaakiaawIcacaGLPaaacaGGSaaa aa@4074@ parce que y MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEaaaa@34F8@ est observé uniquement à la dernière phase d’échantillonnage s p . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4CamaaBa aaleaacaWGWbaabeaakiaac6caaaa@36CF@ Donc, γ ^ i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafq4SdCMbaK aadaWgaaWcbaGaamyAaaqabaaaaa@36CB@ est convergent vers γ i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4SdC2aaS baaSqaaiaadMgaaeqaaaaa@36BB@ pour tout i , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAaiaacY caaaa@3598@ où les β ^ i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqOSdiMbaK aadaWgaaWcbaGaamyAaaqabaaaaa@36C5@ dans γ ^ i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafq4SdCMbaK aadaWgaaWcbaGaamyAaaqabaaaaa@36CB@ sont remplacés par β i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOSdi2aaS baaSqaaiaadMgaaeqaaaaa@36B5@ dans γ i . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4SdC2aaS baaSqaaiaadMgaaeqaaOGaaiOlaaaa@3777@ La convergence n’implique pas nécessairement la convergence des moments et, en particulier, pas de la variance. Cependant, pour une population finie, c’est-à-dire un espace de probabilité fini, les concepts coïncident. Il s’ensuit que, pour n p MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBamaaBa aaleaacaWGWbaabeaaaaa@360E@ suffisamment grand, Var ( Y ^ HT p + i 1 = 1 p ( t ^ i 1 t ^ i 1 + ) γ ^ i 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeOvaiaabg gacaqGYbWaaeWabeaaceWGzbGbaKaadaWgaaWcbaGaaeisaiaabsfa daWgaaadbaGaamiCaaqabaaaleqaaOGaey4kaSYaaabmaeqaleaaca WGPbWaaSbaaWqaaiaaykW7caaIXaaabeaaliaai2dacaaIXaaabaGa amiCaaqdcqGHris5aOWaaeWabeaaceWG0bGbaKaadaqhaaWcbaGaam yAamaaBaaameaacaaMc8UaaGymaaqabaaaleaacqGHsislaaGccqGH sislceWG0bGbaKaadaqhaaWcbaGaamyAamaaBaaameaacaaMc8UaaG ymaaqabaaaleaacqGHRaWkaaaakiaawIcacaGLPaaadaahaaWcbeqa aOGamai2gkdiIcaacuaHZoWzgaqcamaaBaaaleaacaWGPbWaaSbaaW qaaiaaykW7caaIXaaabeaaaSqabaaakiaawIcacaGLPaaaaaa@5B09@ et Var ( Y ^ HT p + i 1 = 1 p ( t ^ i 1 t ^ i 1 + ) γ i 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeOvaiaabg gacaqGYbWaaeWabeaaceWGzbGbaKaadaWgaaWcbaGaaeisaiaabsfa daWgaaadbaGaamiCaaqabaaaleqaaOGaey4kaSYaaabmaeqaleaaca WGPbWaaSbaaWqaaiaaigdaaeqaaSGaaGypaiaaigdaaeaacaWGWbaa niabggHiLdGcdaqadeqaaiqadshagaqcamaaDaaaleaacaWGPbWaaS baaWqaaiaaykW7caaIXaaabeaaaSqaaiabgkHiTaaakiabgkHiTiqa dshagaqcamaaDaaaleaacaWGPbWaaSbaaWqaaiaaykW7caaIXaaabe aaaSqaaiabgUcaRaaaaOGaayjkaiaawMcaamaaCaaaleqabaGccWaG yBOmGikaaiabeo7aNnaaBaaaleaacaWGPbWaaSbaaWqaaiaaykW7ca aIXaaabeaaaSqabaaakiaawIcacaGLPaaaaaa@596E@ sont asymptotiquement équivalents et selon la discussion qui précède, la différence peut être quantifiée par

Var ( w ˜ p  ′ y ) = Var ( Y ^ HT p + r = 1 p ( t ^ r t ^ r + ) γ r ) + N 2 o ( n p 1 ) . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeOvaiaabg gacaqGYbWaaeWaaeaaceWG3bGbaGaadaqhaaWcbaGaamiCaaqaaOGa mai2gkdiIcaacaWG5baacaGLOaGaayzkaaGaaGypaiaabAfacaqGHb GaaeOCamaabmaabaGabmywayaajaWaaSbaaSqaaiaabIeacaqGubWa aSbaaWqaaiaadchaaeqaaaWcbeaakiabgUcaRmaaqahabaWaaeWaae aaceWG0bGbaKaadaqhaaWcbaGaamOCaaqaaiabgkHiTaaakiabgkHi TiqadshagaqcamaaDaaaleaacaWGYbaabaGaey4kaScaaaGccaGLOa GaayzkaaWaaWbaaSqabeaakiadaITHYaIOaaGaeq4SdC2aaSbaaSqa aiaadkhaaeqaaaqaaiaadkhacaaI9aGaaGymaaqaaiaadchaa0Gaey yeIuoaaOGaayjkaiaawMcaaiabgUcaRiaad6eadaahaaWcbeqaaiaa ikdaaaGccaWGVbWaaeWaaeaacaWGUbWaa0baaSqaaiaadchaaeaacq GHsislcaaIXaaaaaGccaGLOaGaayzkaaGaaGOlaaaa@6607@

L’estimateur t ^ r + MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmiDayaaja Waa0baaSqaaiaadkhaaeaacqGHRaWkaaaaaa@3709@ est une sommation sur les unités comprises dans s r , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4CamaaBa aaleaacaWGYbaabeaakiaacYcaaaa@36CF@ tandis que t ^ r MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmiDayaaja Waa0baaSqaaiaadkhaaeaacqGHsislaaaaaa@3714@ est une sommation sur s r 1 . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4CamaaBa aaleaacaWGYbGaeyOeI0IaaGymaaqabaGccaGGUaaaaa@3879@ En réarrangeant les termes, la variance dans le deuxième membre de l’équation peut s’écrire Var ( r = 1 p i S r w r i * e r i ) , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeOvaiaabg gacaqGYbWaaeWaaeaadaaeWaqabSqaaiaadkhacaaI9aGaaGymaaqa aiaadchaa0GaeyyeIuoakmaaqababeWcbaGaamyAaiabgIGiolaado fadaWgaaadbaGaamOCaaqabaaaleqaniabggHiLdGccaWG3bWaa0ba aSqaaiaadkhacaWGPbaabaGaaiOkaaaakiaadwgadaWgaaWcbaGaam OCaiaadMgaaeqaaaGccaGLOaGaayzkaaGaaiilaaaa@4B92@ ce qui est égal à

1 r 1 , r 2 p k U l U w r 1 k * e r 1 k w r 2 l * e r 2 l Cov ( I k s r 1 , I l s r 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaabuaeqale aacaaIXaGaeyizImQaamOCamaaBaaameaacaaMc8UaaGymaaqabaWc caaMb8UaaGilaiaaykW7caWGYbWaaSbaaWqaaiaaykW7caaIYaaabe aaliabgsMiJkaadchaaeqaniabggHiLdGccaaMe8+aaabuaeqaleaa caWGRbGaeyicI4Saamyvaaqab0GaeyyeIuoakiaaysW7daaeqbqaai aadEhadaqhaaWcbaGaamOCamaaBaaameaacaaMc8UaaGymaaqabaWc caWGRbaabaGaaiOkaaaakiaadwgadaWgaaWcbaGaamOCamaaBaaame aacaaMc8UaaGymaaqabaWccaWGRbaabeaakiaadEhadaqhaaWcbaGa amOCamaaBaaameaacaaMc8UaaGOmaaqabaWccaWGSbaabaGaaiOkaa aakiaadwgadaWgaaWcbaGaamOCamaaBaaameaacaaMc8UaaGOmaaqa baWccaWGSbaabeaaaeaacaWGSbGaeyicI4Saamyvaaqab0GaeyyeIu oakiaaboeacaqGVbGaaeODamaabmaabaGaamysamaaBaaaleaacaWG RbGaeyicI4Saam4CamaaBaaameaacaWGYbWaaSbaaeaacaaMc8UaaG ymaaqabaaabeaaaSqabaGccaaMb8UaaiilaiaaysW7caWGjbWaaSba aSqaaiaadYgacqGHiiIZcaWGZbWaaSbaaWqaaiaadkhadaWgaaqaai aaykW7caaIYaaabeaaaeqaaaWcbeaaaOGaayjkaiaawMcaaaaa@839D@

de sorte qu’un estimateur basé sur l’échantillon serait

1 r 1 , r 2 p k s r 1 , l s r 2 w r 1 k * e ^ r 1 k w r 2 l * e ^ r 2 l [ 1 P ( k s r 1 ) P ( l s r 2 ) P ( k s r 1 , l s r 2 ) ] . ( 3.9 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaabuaeqale aacaaIXaGaeyizImQaamOCamaaBaaameaacaaIXaaabeaaliaaygW7 caaISaGaaGjbVlaadkhadaWgaaadbaGaaGOmaaqabaWccqGHKjYOca WGWbaabeqdcqGHris5aOGaaGjbVpaaqafabaGaam4DamaaDaaaleaa caWGYbWaaSbaaWqaaiaaigdaaeqaaSGaam4AaaqaaiaacQcaaaGcce WGLbGbaKaadaWgaaWcbaGaamOCamaaBaaameaacaaIXaaabeaaliaa dUgaaeqaaOGaam4DamaaDaaaleaacaWGYbWaaSbaaWqaaiaaikdaae qaaSGaamiBaaqaaiaacQcaaaGcceWGLbGbaKaadaWgaaWcbaGaamOC amaaBaaameaacaaIYaaabeaaliaadYgaaeqaaaqaaiaadUgacqGHii IZcaWGZbWaaSbaaWqaaiaadkhadaWgaaqaaiaaykW7caaIXaaabeaa aeqaaSGaaGzaVlaaiYcacaaMe8UaamiBaiabgIGiolaadohadaWgaa adbaGaamOCamaaBaaabaGaaGPaVlaaikdaaeqaaaqabaaaleqaniab ggHiLdGcdaWadaqaaiaaigdacqGHsisldaWcaaqaaiaadcfadaqada qaaiaadUgacqGHiiIZcaWGZbWaaSbaaSqaaiaadkhadaWgaaadbaGa aGPaVlaaigdaaeqaaaWcbeaaaOGaayjkaiaawMcaaiaadcfadaqada qaaiaadYgacqGHiiIZcaWGZbWaaSbaaSqaaiaadkhadaWgaaadbaGa aGPaVlaaikdaaeqaaaWcbeaaaOGaayjkaiaawMcaaaqaaiaadcfada qadaqaaiaadUgacqGHiiIZcaWGZbWaaSbaaSqaaiaadkhadaWgaaad baGaaGPaVlaaigdaaeqaaaWcbeaakiaaygW7caaISaGaaGjbVlaadY gacqGHiiIZcaWGZbWaaSbaaSqaaiaadkhadaWgaaadbaGaaGPaVlaa ikdaaeqaaaWcbeaaaOGaayjkaiaawMcaaaaaaiaawUfacaGLDbaaca aIUaGaaGzbVlaaywW7caaMf8UaaGzbVlaaywW7caGGOaGaaG4maiaa c6cacaaI5aGaaiykaaaa@A0E3@

Pour calculer la covariance entre les indicateurs I k s r 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysamaaBa aaleaacaWGRbGaeyicI4Saam4CamaaBaaameaacaWGYbWaaSbaaeaa caaMc8UaaGymaaqabaaabeaaaSqabaaaaa@3BF6@ et I l s r 2 , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysamaaBa aaleaacaWGSbGaeyicI4Saam4CamaaBaaameaacaWGYbWaaSbaaeaa caaMc8UaaGOmaaqabaaabeaaaSqabaGccaGGSaaaaa@3CB2@ nous devons connaître la probabilité conjointe des événements { k s i , l s j } . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiWabeaaca WGRbGaeyicI4Saam4CamaaBaaaleaacaWGPbaabeaakiaaygW7caaI SaGaaGjbVlaadYgacqGHiiIZcaWGZbWaaSbaaSqaaiaadQgaaeqaaa GccaGL7bGaayzFaaGaaiOlaaaa@43CD@ Si s j s i , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4CamaaBa aaleaacaWGQbaabeaakiabgkOimlaadohadaWgaaWcbaGaamyAaaqa baGccaGGSaaaaa@3ADF@ alors P ( k s i , l s j ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiuamaabm qabaGaam4AaiabgIGiolaadohadaWgaaWcbaGaamyAaaqabaGccaaM b8UaaGilaiaaysW7caWGSbGaeyicI4Saam4CamaaBaaaleaacaWGQb aabeaaaOGaayjkaiaawMcaaaaa@4348@ est égale à la probabilité conjointe que les deux unités k , l MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4Aaiaayg W7caaISaGaaGPaVlaadYgaaaa@39A6@ soient dans l’échantillon s i = s min ( i , j ) , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4CamaaBa aaleaacaWGPbaabeaakiaai2dacaWGZbWaaSbaaSqaaiGac2gacaGG PbGaaiOBamaabmaabaGaamyAaiaaygW7caaISaGaaGPaVlaadQgaai aawIcacaGLPaaaaeqaaOGaaGzaVlaacYcaaaa@4448@ multipliée par la probabilité conditionnelle que l’unité l MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiBaaaa@34EB@ soit dans l’échantillon s j , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4CamaaBa aaleaacaWGQbaabeaakiaacYcaaaa@36C7@ sachant qu’il appartient à s i . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4CamaaBa aaleaacaWGPbaabeaakiaac6caaaa@36C8@ Formellement, si s j s i , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4CamaaBa aaleaacaWGQbaabeaakiabgkOimlaadohadaWgaaWcbaGaamyAaaqa baGccaGGSaaaaa@3ADF@ alors P ( k s i , l s j ) = w i l * w j l * w i , l k * 1 , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiuamaabm qabaGaam4AaiabgIGiolaadohadaWgaaWcbaGaamyAaaqabaGccaaM b8UaaGilaiaaysW7caWGSbGaeyicI4Saam4CamaaBaaaleaacaWGQb aabeaaaOGaayjkaiaawMcaaiaai2dadaWcbaWcbaGaam4DamaaDaaa meaacaWGPbGaamiBaaqaaiaacQcaaaaaleaacaWG3bWaa0baaWqaai aadQgacaWGSbaabaGaaiOkaaaaaaGccaWG3bWaa0baaSqaaiaadMga caaMb8UaaGilaiaaykW7caWGSbGaam4AaaqaaiaacQcacqGHsislca aIXaaaaOGaaiilaaaa@5682@ ce qui élimine la dépendance à l’égard de s r 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4CamaaBa aaleaacaWGYbWaaSbaaeaacaaIYaaabeaaaeqaaaaa@36F1@ entre les crochets dans (3.9) et le résultat s’ensuit.

Un autre moyen d’écrire (3.8) est

1 r p k , l s r ( w r k * w r l * w r k l * ) e ^ r k e ^ r l + 2 1 r m < r M p k s r m l s r M w r m k * e ^ r m k w r M l * e ^ r M l ( 1 w r m k l * w r m k * w r m l * ) . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaabuaeaaca aMc8+aaabuaeaadaqadaqaaiaadEhadaqhaaWcbaGaamOCaiaadUga aeaacaGGQaaaaOGaam4DamaaDaaaleaacaWGYbGaamiBaaqaaiaacQ caaaGccqGHsislcaWG3bWaa0baaSqaaiaadkhacaWGRbGaamiBaaqa aiaacQcaaaaakiaawIcacaGLPaaaceWGLbGbaKaadaWgaaWcbaGaam OCaiaadUgaaeqaaOGabmyzayaajaWaaSbaaSqaaiaadkhacaWGSbaa beaaaeaacaWGRbGaaGzaVlaaiYcacaaMc8UaamiBaiabgIGiolaado hadaWgaaadbaGaamOCaaqabaaaleqaniabggHiLdaaleaacaaIXaGa eyizImQaamOCaiabgsMiJkaadchaaeqaniabggHiLdGccqGHRaWkca aIYaWaaabuaeqaleaacaaIXaGaeyizImQaamOCamaaBaaameaacaWG TbaabeaaliaaiYdacaWGYbWaaSbaaWqaaiaad2eaaeqaaSGaeyizIm QaamiCaaqab0GaeyyeIuoakiaaysW7daaeqbqabSqaaiaadUgacqGH iiIZcaWGZbWaaSbaaWqaaiaadkhadaWgaaqaaiaad2gaaeqaaaqaba aaleqaniabggHiLdGccaaMe8+aaabuaeaacaWG3bWaa0baaSqaaiaa dkhadaWgaaadbaGaamyBaaqabaWccaWGRbaabaGaaiOkaaaakiqadw gagaqcamaaBaaaleaacaWGYbWaaSbaaWqaaiaad2gaaeqaaSGaam4A aaqabaGccaWG3bWaa0baaSqaaiaadkhadaWgaaadbaGaamytaaqaba WccaWGSbaabaGaaiOkaaaakiqadwgagaqcamaaBaaaleaacaWGYbWa aSbaaWqaaiaad2eaaeqaaSGaamiBaaqabaaabaGaamiBaiabgIGiol aadohadaWgaaadbaGaamOCamaaBaaabaGaamytaaqabaaabeaaaSqa b0GaeyyeIuoakmaabmaabaGaaGymaiabgkHiTmaalaaabaGaam4Dam aaDaaaleaacaWGYbWaaSbaaWqaaiaad2gaaeqaaSGaam4AaiaadYga aeaacaGGQaaaaaGcbaGaam4DamaaDaaaleaacaWGYbWaaSbaaWqaai aad2gaaeqaaSGaam4AaaqaaiaacQcaaaGccaWG3bWaa0baaSqaaiaa dkhadaWgaaadbaGaamyBaaqabaWccaWGSbaabaGaaiOkaaaaaaaaki aawIcacaGLPaaacaaIUaaaaa@A46A@

Quand p = 2 , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCaiaai2 dacaaIYaGaaiilaaaa@3722@ les termes γ i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4SdC2aaS baaSqaaiaadMgaaeqaaaaa@36BB@ coïncident avec les unités de variation obtenues de la décomposition de l’erreur d’échantillonnage de l’estimateur en deux étapes de Breidt et Fuller (1993). Des estimations convergentes pour les écarts-types des estimations calées des sous-totaux de population sont calculées de façon ordinaire en multipliant la variable cible par une variable indicatrice pour la sous-population particulière.

Jusqu’à présent dans notre discussion, nous avons donné une représentation du vecteur de poids calés de laquelle nous avons dérivé un nouvel estimateur convergent pour la variance des estimateurs calés en plusieurs phases. Cependant, dans certains cas, les estimateurs peuvent être simplifiés davantage sans perte d’exactitude. Nous discuterons brièvement ici de deux scénarios qui dépendent du fait que n j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBamaaBa aaleaacaWGQbaabeaaaaa@3608@ est ou non significativement plus petit que n j 1 , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBamaaBa aaleaacaWGQbGaeyOeI0IaaGymaaqabaGccaGGSaaaaa@386A@ c’est-à-dire du fait que, pour tout j , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOAaiaacY caaaa@3599@ le sous-échantillon s j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4CamaaBa aaleaacaWGQbaabeaaaaa@360D@ est ou non significativement plus petit que s j 1 . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4CamaaBa aaleaacaWGQbGaeyOeI0IaaGymaaqabaGccaGGUaaaaa@3871@ Un cas type du premier scénario est celui où l’on possède un ensemble de fichiers administratifs emboîtés dont les tailles diminuent significativement. Le premier ensemble peut être, par exemple, un fichier de registre de population qui contient un nombre limité de variables au sujet de l’ensemble de la population, comme l’âge, le sexe, etc. Le deuxième ensemble peut correspondre à des données d’échantillons provenant d’une enquête de portée nationale dans le cadre de laquelle des données complètes sur les ménages ont été recueillies auprès de toutes les unités échantillonnées, mais en utilisant un questionnaire supplémentaire pour un sous-groupe de ces unités (disons, une unité sur dix). Les données pour ce sous-groupe d’unités peuvent alors être calées sur celles provenant des deux sources d’information précédentes. Un exemple du second scénario est la situation où une ou deux phases de calage sont effectuées sur le même ensemble de données. Autrement dit, contrairement au processus à plusieurs phases habituel, l’élément d’échantillonnage est présent à la première phase seulement, mais non aux phases ultérieures. Un tel scénario peut avoir lieu si nous voulons caler les données d’une enquête sur de nombreuses variables pour lesquelles nous connaissons seulement les totaux de marge, mais ne possédons pas les totaux transversaux. Dans ces conditions, une série de calages sur le même échantillon, mais en utilisant un ensemble différent de variables auxiliaires à chaque phase, en attribuant habituellement aux dernières phases les variables les plus importantes, pourrait être un compromis satisfaisant. Une meilleure façon de caractériser ce scénario serait de le dire séquentiel. Sous ces scénarios, w ˜ p MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabm4Dayaaia WaaSbaaSqaaiaadchaaeqaaaaa@3626@ et sa variance peuvent être simplifiés considérablement. Ces scénarios peuvent être énoncés comme des corollaires de notre analyse, mais nous choisissons de ne pas les prendre en considération ici afin de nous concentrer sur nos résultats courants.

3.2 Exemples : Calage à deux phases et à trois phases

Calage à deux phases. Nous utiliserons le cas particulier du calage à deux phases ( p = 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaaca WGWbGaaGypaiaaikdaaiaawIcacaGLPaaaaaa@37FB@ pour démontrer la nouvelle méthodologie et ce qui la distingue de l’autre estimateur habituellement utilisé dans la littérature. En notation matricielle, l’estimateur calé est donné, selon (3.7), par

w ˜ 2  ′ y = Y ^ HT 2 + ( t ^ 1 t ^ 1 + ) γ ^ 1 + ( t ^ 2 t ^ 2 + ) γ ^ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabm4Dayaaia Waa0baaSqaaiaaikdaaeaakiadaITHYaIOaaGaamyEaiaai2daceWG zbGbaKaadaWgaaWcbaGaaeisaiaabsfadaWgaaadbaGaaGOmaaqaba aaleqaaOGaey4kaSYaaeWaaeaaceWG0bGbaKaadaqhaaWcbaGaaGym aaqaaiabgkHiTaaakiabgkHiTiqadshagaqcamaaDaaaleaacaaIXa aabaGaey4kaScaaaGccaGLOaGaayzkaaWaaWbaaSqabeaakiadaITH YaIOaaGafq4SdCMbaKaadaWgaaWcbaGaaGymaaqabaGccqGHRaWkda qadaqaaiqadshagaqcamaaDaaaleaacaaIYaaabaGaeyOeI0caaOGa eyOeI0IabmiDayaajaWaa0baaSqaaiaaikdaaeaacqGHRaWkaaaaki aawIcacaGLPaaadaahaaWcbeqaaOGamai2gkdiIcaacuaHZoWzgaqc amaaBaaaleaacaaIYaaabeaaaaa@5C07@

γ ^ 1 = β ^ 1 Z ^ 12 β ^ 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafq4SdCMbaK aadaWgaaWcbaGaaGymaaqabaGccaaI9aGafqOSdiMbaKaadaWgaaWc baGaaGymaaqabaGccqGHsislceWGAbGbaKaadaWgaaWcbaGaaGymai aaikdaaeqaaOGafqOSdiMbaKaadaWgaaWcbaGaaGOmaaqabaaaaa@402D@ et γ ^ 2 = β ^ 2 . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafq4SdCMbaK aadaWgaaWcbaGaaGOmaaqabaGccaaI9aGafqOSdiMbaKaadaWgaaWc baGaaGOmaaqabaGccaGGUaaaaa@3ABF@ Explicitement, sous forme non matricielle,

w ˜ 2  ′ y = k s 2 w 2 k * y k + ( k U x 1 k k s 1 w 1 k x 1 k ) γ ^ 1 + ( k s 1 w 1 k x 2 k k s 2 w 2 k * x 2 k ) γ ^ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabm4Dayaaia Waa0baaSqaaiaaikdaaeaakiadaITHYaIOaaGaamyEaiaai2dadaae qbqaaiaadEhadaqhaaWcbaGaaGOmaiaadUgaaeaacaGGQaaaaOGaam yEamaaBaaaleaacaWGRbaabeaaaeaacaWGRbGaeyicI4Saam4Camaa BaaameaacaaIYaaabeaaaSqab0GaeyyeIuoakiabgUcaRmaabmaaba WaaabuaeqaleaacaWGRbGaeyicI4Saamyvaaqab0GaeyyeIuoakiaa dIhadaWgaaWcbaGaaGymaiaadUgaaeqaaOGaeyOeI0Yaaabuaeqale aacaWGRbGaeyicI4Saam4CamaaBaaameaacaaIXaaabeaaaSqab0Ga eyyeIuoakiaadEhadaWgaaWcbaGaaGymaiaadUgaaeqaaOGaamiEam aaBaaaleaacaaIXaGaam4AaaqabaaakiaawIcacaGLPaaacuaHZoWz gaqcamaaBaaaleaacaaIXaaabeaakiabgUcaRmaabmaabaWaaabuae aacaWG3bWaaSbaaSqaaiaaigdacaWGRbaabeaakiaadIhadaWgaaWc baGaaGOmaiaadUgaaeqaaaqaaiaadUgacqGHiiIZcaWGZbWaaSbaaW qaaiaaigdaaeqaaaWcbeqdcqGHris5aOGaeyOeI0YaaabuaeaacaWG 3bWaa0baaSqaaiaaikdacaWGRbaabaGaaiOkaaaakiaadIhadaWgaa WcbaGaaGOmaiaadUgaaeqaaaqaaiaadUgacqGHiiIZcaWGZbWaaSba aWqaaiaaikdaaeqaaaWcbeqdcqGHris5aaGccaGLOaGaayzkaaGafq 4SdCMbaKaadaWgaaWcbaGaaGOmaaqabaaaaa@8075@

γ ^ 1 = ( k s 1 w 1 k x 1 k x 1 k ) 1 [ k s 2 w 2 k * x 1 k y k ( k s 2 w 2 k * x 1 k x 2 k k s 1 w 1 k x 1 k x 2 k ) γ ^ 2 ] γ ^ 2 = ( k s 2 w 2 k * x 2 k x 2 k ) 1 k s 2 w 2 k * x 2 k y k . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFjpu0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqbaeaabiGaaa qaaiqbeo7aNzaajaWaaSbaaSqaaiaaigdaaeqaaaGcbaGaaGypamaa bmaabaWaaabuaeaacaWG3bWaaSbaaSqaaiaaigdacaWGRbaabeaaki aadIhadaWgaaWcbaGaaGymaiaadUgaaeqaaOGaamiEamaaDaaaleaa caaIXaGaam4AaaqaaOGamai2gkdiIcaaaSqaaiaadUgacqGHiiIZca WGZbWaaSbaaWqaaiaaigdaaeqaaaWcbeqdcqGHris5aaGccaGLOaGa ayzkaaWaaWbaaSqabeaacqGHsislcaaIXaaaaOWaamWaaeaadaaeqb qaaiaadEhadaqhaaWcbaGaaGOmaiaadUgaaeaacaGGQaaaaOGaamiE amaaBaaaleaacaaIXaGaam4AaaqabaGccaWG5bWaaSbaaSqaaiaadU gaaeqaaaqaaiaadUgacqGHiiIZcaWGZbWaaSbaaWqaaiaaikdaaeqa aaWcbeqdcqGHris5aOGaeyOeI0YaaeWaaeaadaaeqbqaaiaadEhada qhaaWcbaGaaGOmaiaadUgaaeaacaGGQaaaaOGaamiEamaaBaaaleaa caaIXaGaam4AaaqabaGccaWG4bWaa0baaSqaaiaaikdacaWGRbaaba GccWaGyBOmGikaaaWcbaGaam4AaiabgIGiolaadohadaWgaaadbaGa aGOmaaqabaaaleqaniabggHiLdGccqGHsisldaaeqbqaaiaadEhada WgaaWcbaGaaGymaiaadUgaaeqaaOGaamiEamaaBaaaleaacaaIXaGa am4AaaqabaGccaWG4bWaa0baaSqaaiaaikdacaWGRbaabaGccWaGyB OmGikaaaWcbaGaam4AaiabgIGiolaadohadaWgaaadbaGaaGymaaqa baaaleqaniabggHiLdaakiaawIcacaGLPaaacuaHZoWzgaqcamaaBa aaleaacaaIYaaabeaaaOGaay5waiaaw2faaaqaaiqbeo7aNzaajaWa aSbaaSqaaiaaikdaaeqaaaGcbaGaaGypamaabmaabaWaaabuaeaaca WG3bWaa0baaSqaaiaaikdacaWGRbaabaGaaiOkaaaakiaadIhadaWg aaWcbaGaaGOmaiaadUgaaeqaaOGaamiEamaaDaaaleaacaaIYaGaam 4AaaqaaOGamai2gkdiIcaaaSqaaiaadUgacqGHiiIZcaWGZbWaaSba aWqaaiaaikdaaeqaaaWcbeqdcqGHris5aaGccaGLOaGaayzkaaWaaW baaSqabeaacqGHsislcaaIXaaaaOWaaabuaeaacaWG3bWaa0baaSqa aiaaikdacaWGRbaabaGaaiOkaaaakiaadIhadaWgaaWcbaGaaGOmai aadUgaaeqaaOGaamyEamaaBaaaleaacaWGRbaabeaakiaaygW7caGG UaaaleaacaWGRbGaeyicI4Saam4CamaaBaaameaacaaIYaaabeaaaS qab0GaeyyeIuoaaaaaaa@B349@

Cet estimateur produit des estimations identiques à l’estimateur calé en deux phases utilisé dans Hidiroglou et Särndal (1998) ou dans Särndal et coll. (1992), section 9.7. Cependant, une fois que l’estimateur des paramètres γ 1 , γ 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4SdC2aaS baaSqaaiaaigdaaeqaaOGaaGzaVlaaiYcacaaMe8Uaeq4SdC2aaSba aSqaaiaaikdaaeqaaaaa@3CEE@ est calculé, la représentation de w ˜ 2  ′ y MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabm4Dayaaia Waa0baaSqaaiaaikdaaeaakiadaITHYaIOaaGaamyEaaaa@39D6@ devient simple et informative, car elle possède la structure d’un simple estimateur par la régression multivariée. Cet estimateur linéaire est fondé sur les coefficients γ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4SdCgaaa@35A1@ qui englobent l’effet total de la variable x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCiEaaaa@34FB@ qu’ils multiplient et, donc, diffèrent légèrement des coefficients β . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOSdiMaai Olaaaa@364D@ γ ^ i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafq4SdCMbaK aadaWgaaWcbaGaamyAaaqabaaaaa@36CB@ englobe l’effet global que le calage sur la variable x i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCiEamaaBa aaleaacaWGPbaabeaaaaa@3615@ a sur l’estimation de Y . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamywaiaac6 caaaa@358A@ Dans le cas général, il tient compte de la projection de y MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEaaaa@34F8@ sur x i , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCiEamaaBa aaleaacaWGPbaabeaakiaacYcaaaa@36CF@ de la projection de y MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEaaaa@34F8@ sur x i + 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCiEamaaBa aaleaacaWGPbGaey4kaSIaaGymaaqabaaaaa@37B2@ multipliée par la projection de x i + 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCiEamaaBa aaleaacaWGPbGaey4kaSIaaGymaaqabaaaaa@37B2@ sur x i , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCiEamaaBa aaleaacaWGPbaabeaakiaacYcaaaa@36CF@ et ainsi de suite. En outre, comme nous allons le montrer, les estimateurs de variance diffèrent significativement en ce qui concerne tant les estimations que la représentation. Étant donné la complexité de l’évaluation de la variance des estimateurs qui comprennent des facteurs g , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4zaiaacY caaaa@35B6@ jusqu’à présent dans la littérature sur le calage à deux phases, il était d’usage pratique de commencer par donner aux facteurs g MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4zaaaa@3506@ la valeur approximative de 1, puis d’utiliser la loi de la variation totale pour obtenir deux composantes, une pour chaque phase, conformément à

V ^ C ( w ˜ 2  ′ y ) = k , l s 2 w 2 k l ( w 1 k w 1 l w 1 k l ) ( g 1 k e 1 k ) ( g 1 l e 1 l ) + k , l s 2 w 1 k w 1 l ( w 2 k w 2 l w 2 k l ) ( g 2 k e 2 k ) ( g 2 l e 2 l ) ( 3.10 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=jFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqbaeaabiGaaa qaaiqadAfagaqcamaaBaaaleaacaWGdbaabeaakmaabmaabaGabm4D ayaaiaWaa0baaSqaaiaaikdaaeaakiadaITHYaIOaaGaamyEaaGaay jkaiaawMcaaaqaaiaai2dadaaeqbqabSqaaiaadUgacaaMb8UaaGil aiaaykW7caWGSbGaeyicI4Saam4CamaaBaaameaacaaIYaaabeaaaS qab0GaeyyeIuoakiaadEhadaWgaaWcbaGaaGOmaiaadUgacaWGSbaa beaakmaabmaabaGaam4DamaaBaaaleaacaaIXaGaam4AaaqabaGcca WG3bWaaSbaaSqaaiaaigdacaWGSbaabeaakiabgkHiTiaadEhadaWg aaWcbaGaaGymaiaadUgacaWGSbaabeaaaOGaayjkaiaawMcaamaabm aabaGaam4zamaaBaaaleaacaaIXaGaam4AaaqabaGcceWGLbGbaqba daWgaaWcbaGaaGymaiaadUgaaeqaaaGccaGLOaGaayzkaaWaaeWaae aacaWGNbWaaSbaaSqaaiaaigdacaWGSbaabeaakiqadwgagaafamaa BaaaleaacaaIXaGaamiBaaqabaaakiaawIcacaGLPaaaaeaaaeaaca aMc8UaaGPaVlabgUcaRmaaqafabeWcbaGaam4AaiaaygW7caaISaGa aGPaVlaadYgacqGHiiIZcaWGZbWaaSbaaWqaaiaaikdaaeqaaaWcbe qdcqGHris5aOGaam4DamaaBaaaleaacaaIXaGaam4AaaqabaGccaWG 3bWaaSbaaSqaaiaaigdacaWGSbaabeaakmaabmaabaGaam4DamaaBa aaleaacaaIYaGaam4AaaqabaGccaWG3bWaaSbaaSqaaiaaikdacaWG SbaabeaakiabgkHiTiaadEhadaWgaaWcbaGaaGOmaiaadUgacaWGSb aabeaaaOGaayjkaiaawMcaamaabmaabaGaam4zamaaBaaaleaacaaI YaGaam4AaaqabaGcceWGLbGbaqbadaWgaaWcbaGaaGOmaiaadUgaae qaaaGccaGLOaGaayzkaaWaaeWaaeaacaWGNbWaaSbaaSqaaiaaikda caWGSbaabeaakiqadwgagaafamaaBaaaleaacaaIYaGaamiBaaqaba aakiaawIcacaGLPaaacaaMf8UaaGzbVlaaywW7caaMf8UaaGzbVlaa cIcacaaIZaGaaiOlaiaaigdacaaIWaGaaiykaaaaaaa@A341@

où les termes d’erreur e 1 k = y k x 1 k γ ^ 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmyzayaaua WaaSbaaSqaaiaaigdacaWGRbaabeaakiaai2dacaWG5bWaaSbaaSqa aiaadUgaaeqaaOGaeyOeI0IaamiEamaaDaaaleaacaaIXaGaam4Aaa qaaOGamai2gkdiIcaacuaHZoWzgaqcamaaBaaaleaacaaIXaaabeaa aaa@4315@ et e 2 k = y k x 2 k γ ^ 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmyzayaaua WaaSbaaSqaaiaaikdacaWGRbaabeaakiaai2dacaWG5bWaaSbaaSqa aiaadUgaaeqaaOGaeyOeI0IaamiEamaaDaaaleaacaaIYaGaam4Aaa qaaOGamai2gkdiIcaacuaHZoWzgaqcamaaBaaaleaacaaIYaaabeaa aaa@4318@ sont tous deux définis pour k s 2 , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4AaiabgI GiolaadohadaWgaaWcbaGaaGOmaaqabaGccaaMb8Uaaiilaaaa@3A92@ parce que y MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEaaaa@34F8@ est observé uniquement sur s 2 , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4CamaaBa aaleaacaaIYaaabeaakiaacYcaaaa@3694@ et on notera la représentation simple des termes d’erreur sous la notation faisant appel aux coefficients γ . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4SdCMaai Olaaaa@3653@ Les facteurs g MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4zaaaa@3506@ sont définis comme dans (3.5). La valeur approximative de 1 donnée aux facteurs g MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4zaaaa@3506@ dans le calcul de (3.10) peut indubitablement aboutir à des estimations imprévisibles, car ces facteurs s’écartent de l’unité précisément dans les situations où le calage est essentiel. Par ailleurs, l’estimateur de variance proposé en (3.8) pour un estimateur calé en deux phases est donné par

V ^ P ( w ˜ 2  ′ y ) = k , l s 1 ( w 1 k w 1 l w 1 k l ) e ^ 1 k e ^ 1 l + k , l s 2 ( w 2 k * w 2 l * w 2 k l * ) e ^ 2 k e ^ 2 l + 2 k s 1 , l s 2 w 2 l * w 1 l ( w 1 k w 1 l w 1 k l ) e ^ 1 k e ^ 2 l . ( 3.11 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=jFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqbaeaabiGaaa qaaiqadAfagaqcamaaBaaaleaacaWGqbaabeaakmaabmaabaGabm4D ayaaiaWaa0baaSqaaiaaikdaaeaakiadaITHYaIOaaGaamyEaaGaay jkaiaawMcaaaqaaiaai2dadaaeqbqabSqaaiaadUgacaaMb8UaaGil aiaaysW7caWGSbGaeyicI4Saam4CamaaBaaameaacaaIXaaabeaaaS qab0GaeyyeIuoakmaabmaabaGaam4DamaaBaaaleaacaaIXaGaam4A aaqabaGccaWG3bWaaSbaaSqaaiaaigdacaWGSbaabeaakiabgkHiTi aadEhadaWgaaWcbaGaaGymaiaadUgacaWGSbaabeaaaOGaayjkaiaa wMcaaiqadwgagaqcamaaBaaaleaacaaIXaGaam4AaaqabaGcceWGLb GbaKaadaWgaaWcbaGaaGymaiaadYgaaeqaaOGaey4kaSYaaabuaeqa leaacaWGRbGaaGilaiaadYgacqGHiiIZcaWGZbWaaSbaaWqaaiaaik daaeqaaaWcbeqdcqGHris5aOWaaeWaaeaacaWG3bWaa0baaSqaaiaa ikdacaWGRbaabaGaaiOkaaaakiaadEhadaqhaaWcbaGaaGOmaiaadY gaaeaacaGGQaaaaOGaeyOeI0Iaam4DamaaDaaaleaacaaIYaGaam4A aiaadYgaaeaacaGGQaaaaaGccaGLOaGaayzkaaGabmyzayaajaWaaS baaSqaaiaaikdacaWGRbaabeaakiqadwgagaqcamaaBaaaleaacaaI YaGaamiBaaqabaaakeaaaeaacaaMc8UaaGPaVlabgUcaRiaaikdada aeqbqabSqaaiaadUgacqGHiiIZcaWGZbWaaSbaaWqaaiaaigdaaeqa aSGaaGzaVlaaiYcacaaMe8UaamiBaiabgIGiolaadohadaWgaaadba GaaGOmaaqabaaaleqaniabggHiLdGcdaWcaaqaaiaadEhadaqhaaWc baGaaGOmaiaadYgaaeaacaGGQaaaaaGcbaGaam4DamaaBaaaleaaca aIXaGaamiBaaqabaaaaOWaaeWaaeaacaWG3bWaaSbaaSqaaiaaigda caWGRbaabeaakiaadEhadaWgaaWcbaGaaGymaiaadYgaaeqaaOGaey OeI0Iaam4DamaaBaaaleaacaaIXaGaam4AaiaadYgaaeqaaaGccaGL OaGaayzkaaGabmyzayaajaWaaSbaaSqaaiaaigdacaWGRbaabeaaki qadwgagaqcamaaBaaaleaacaaIYaGaamiBaaqabaGccaaIUaGaaGzb VlaaywW7caaMf8UaaGzbVlaaywW7caaMf8UaaGzbVlaaywW7caaMf8 UaaGzbVlaacIcacaaIZaGaaiOlaiaaigdacaaIXaGaaiykaaaaaaa@B840@

La différence entre les estimateurs de variance issus des deux méthodes représentées par les équations (3.10) et (3.11) est fondamentale. Elle se manifeste sous divers aspects. Tandis que le terme d’erreur de la deuxième phase est le même dans les deux méthodes, c’est-à-dire e ^ 2 k = e 2 k , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmyzayaaja WaaSbaaSqaaiaaikdacaWGRbaabeaakiaai2daceWGLbGbaqbadaWg aaWcbaGaaGOmaiaadUgaaeqaaOGaaiilaaaa@3B34@ le terme d’erreur de la première phase diffère. e 1 k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmyzayaaua WaaSbaaSqaaiaaigdacaWGRbaabeaaaaa@36D6@ est fondé sur la différence entre y k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEamaaBa aaleaacaWGRbaabeaaaaa@3614@ et le prédicteur de régression x 1 k γ ^ 1 , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaDa aaleaacaaIXaGaam4AaaqaaOGamai2gkdiIcaacuaHZoWzgaqcamaa BaaaleaacaaIXaaabeaakiaaygW7caGGSaaaaa@3E9B@ tandis que e ^ 1 k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmyzayaaja WaaSbaaSqaaiaaigdacaWGRbaabeaaaaa@36CB@ est basé sur la différence entre deux prédicteurs de Y MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamywaaaa@34D8@ provenant des phases un et deux x 1 k γ ^ 1 x 2 k γ ^ 2 . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaDa aaleaacaaIXaGaam4AaaqaaOGamai2gkdiIcaacuaHZoWzgaqcamaa BaaaleaacaaIXaaabeaakiabgkHiTiaadIhadaqhaaWcbaGaaGOmai aadUgaaeaakiadaITHYaIOaaGafq4SdCMbaKaadaWgaaWcbaGaaGOm aaqabaGccaaMb8UaaiOlaaaa@47F3@ Cette modification fait que le premier opérande dans (3.11) est calculé sur s 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4CamaaBa aaleaacaaIXaaabeaaaaa@35D9@ et non sur s 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4CamaaBa aaleaacaaIYaaabeaaaaa@35DA@ où l’échantillon est plus grand. Comme on le voit, l’estimateur (3.11) comprend un troisième opérande qui contient le produit des deux termes d’erreur provenant des deux phases et n’a pas de parallèle dans (3.10). Bien que ce produit soit souvent proche de zéro quand les termes d’erreur ne sont pas fortement corrélés, il peut être non négligeable quand y MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEaaaa@34F8@ est fortement corrélé avec x 1 . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCiEamaaBa aaleaacaaIXaaabeaakiaac6caaaa@369E@ Un avantage évident est l’absence des facteurs g MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4zaaaa@3506@ qui rend l’estimateur plus simple à calculer, c’est-à-dire qu’une fois que nous avons calculé les estimations des paramètres γ ^ i ; i = 1 p , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafq4SdCMbaK aadaWgaaWcbaGaamyAaaqabaGccaaMb8UaaG4oaiaaysW7caWGPbGa aGypaiaaigdacqWIMaYscaWGWbGaaiilaaaa@3FE8@ l’estimateur (3.11) peut être calculé en utilisant les paramètres du plan uniquement, sans impliquer les facteurs g MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4zaaaa@3506@ provenant de toutes les phases du calage. Enfin, aspect peut-être le plus important du point de vue opérationnel, comme nous le montrerons aussi dans l’étude en simulation, l’avantage de (3.11) est que, pour une grande gamme de plans de sondage, le deuxième opérande représente la majorité absolue de la variance, tandis que dans (3.10), les opérandes sont habituellement du même ordre de grandeur. Cette caractéristique découle du fait que le terme ( w 2 k * w 2 l * w 2 k l * ) , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaaca WG3bWaa0baaSqaaiaaikdacaWGRbaabaGaaiOkaaaakiaadEhadaqh aaWcbaGaaGOmaiaadYgaaeaacaGGQaaaaOGaeyOeI0Iaam4DamaaDa aaleaacaaIYaGaam4AaiaadYgaaeaacaGGQaaaaaGccaGLOaGaayzk aaGaaiilaaaa@42B9@ qui comprend les poids d’échantillonnage totaux, est très grand comparativement à w 2 k l ( w 1 k w 1 l w 1 k l ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4DamaaBa aaleaacaaIYaGaam4AaiaadYgaaeqaaOWaaeWaaeaacaWG3bWaaSba aSqaaiaaigdacaWGRbaabeaakiaadEhadaWgaaWcbaGaaGymaiaadY gaaeqaaOGaeyOeI0Iaam4DamaaBaaaleaacaaIXaGaam4AaiaadYga aeqaaaGccaGLOaGaayzkaaaaaa@43C8@ ou w 1 k w 1 l ( w 2 k w 2 l w 2 k l ) . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4DamaaBa aaleaacaaIXaGaam4AaaqabaGccaWG3bWaaSbaaSqaaiaaigdacaWG SbaabeaakmaabmaabaGaam4DamaaBaaaleaacaaIYaGaam4Aaaqaba GccaWG3bWaaSbaaSqaaiaaikdacaWGSbaabeaakiabgkHiTiaadEha daWgaaWcbaGaaGOmaiaadUgacaWGSbaabeaaaOGaayjkaiaawMcaai aac6caaaa@4669@ Dans l’estimateur de variance, la fonction f ( w ) = w k w l w k l MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaabm aabaGaam4DaaGaayjkaiaawMcaaiaai2dacaWG3bWaaSbaaSqaaiaa dUgaaeqaaOGaam4DamaaBaaaleaacaWGSbaabeaakiabgkHiTiaadE hadaWgaaWcbaGaam4AaiaadYgaaeqaaaaa@406C@ atteint son maximum sur la diagonale k = l , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4Aaiaai2 dacaWGSbGaaiilaaaa@3752@ où elle est proportionnelle à w k 2 , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4DamaaDa aaleaacaWGRbaabaGaaGOmaaaakiaacYcaaaa@3789@ et puis elle est multipliée par le carré de son reste e ^ k , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmyzayaaja WaaSbaaSqaaiaadUgaaeqaaOGaaiilaaaa@36CA@ qui est un terme non négatif. D’où, quand le taux d’échantillonnage de la seconde phase est suffisamment élevé, il accroît fortement les termes qui dépendent des poids totaux w 2 * MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4DamaaDa aaleaacaaIYaaabaGaaiOkaaaaaaa@368D@ de cette phase, comparativement à un terme parallèle provenant de la phase précédente. Donc, le deuxième opérande peut, pratiquement à lui seul, être un bon estimateur de la variance de l’estimateur calé.

Calage à trois phases. Le calage à plusieurs phases peut être mis en œuvre quand, dans une série d’échantillons de taille décroissante (non croissante), chaque paire de phases consécutives présente certaines variables communes. Il peut être effectué que les échantillons soient emboîtés, c’est-à-dire si s i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4CamaaBa aaleaacaWGPbaabeaaaaa@360C@ est un sous-échantillon de s i 1 , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4CamaaBa aaleaacaWGPbGaeyOeI0IaaGymaaqabaGccaGGSaaaaa@386E@ ou non. En pratique, le cas le plus simple et le plus fréquent est évidemment le calage à deux phases où un plus petit échantillon (emboîté ou non) est calé sur un échantillon beaucoup plus grand, comme celui d’une Enquête sur la population active, qui est à son tour fréquemment calé sur un fichier administratif contenant des variables démographiques. Cependant, étant donné la faisabilité des calculs et les progrès méthodologiques, les plans comportant un plus grand nombre de phases de calage demeurent répandus et les plans à trois phases occupent le second rang quant à la simplicité et à la mise en œuvre. Par conséquent, cela vaut la peine de s’étendre un peu plus sur l’estimateur pour ce cas.

L’approximation (3.8) contient six termes différents, trois pour les trois phases d’échantillonnage et trois autres pour la covariance entre les phases. Nous désignons ces termes par V 1 , V 2 , V 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOvamaaBa aaleaacaaIXaaabeaakiaaygW7caaISaGaaGjbVlaadAfadaWgaaWc baGaaGOmaaqabaGccaaMb8UaaGilaiaaysW7caWGwbWaaSbaaSqaai aaiodaaeqaaaaa@40F1@ et C 12 , C 13 , C 23 , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4qamaaBa aaleaacaaIXaGaaGOmaaqabaGccaaMb8UaaGilaiaaysW7caWGdbWa aSbaaSqaaiaaigdacaaIZaaabeaakiaaygW7caaISaGaaGjbVlaado eadaWgaaWcbaGaaGOmaiaaiodaaeqaaOGaaGzaVlaacYcaaaa@4530@ respectivement. Chacun correspond à la multiplication d’un terme qui comprend les poids d’échantillonnage par les restes pour les phases pertinentes. Les formules pour le calage à trois phases sont présentées à l’annexe B. Comme nous l’avons exposé pour le cas à deux phases, quand w i > 1 , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4DamaaBa aaleaacaWGPbaabeaakiaai6dacaaIXaGaaiilaaaa@384D@ les V i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOvamaaBa aaleaacaWGPbaabeaaaaa@35EF@ suivent vraisemblablement un ordre clair V 1 < V 2 < V 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOvamaaBa aaleaacaaIXaaabeaakiaaiYdacaWGwbWaaSbaaSqaaiaaikdaaeqa aOGaaGipaiaadAfadaWgaaWcbaGaaG4maaqabaaaaa@3AE3@ et V 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOvamaaBa aaleaacaaIZaaabeaaaaa@35BE@ deviendra d’autant plus dominant que les taux d’échantillonnage de la troisième phase seront grands. Cette situation est représentée par le cas 3 dans le tableau 3.1, et dans notre simulation, cela se manifeste aux lignes 2 et 6 du tableau 4.2, où w 3 k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4DamaaBa aaleaacaaIZaGaam4Aaaqabaaaaa@36CF@ est égal à 10 et à 5, respectivement. Ce n’est manifestement pas très souvent le cas en réalité, car l’approximation dépend aussi des tailles des termes de reste, qui dépendent du choix des variables de calage et de leurs corrélations particulières qui sont parfois très fortes. Le cas échéant, les restes seront très petits et il serait préférable d’utiliser tous les termes de (3.8). Comme pour les termes de covariance, même si C 13 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4qamaaBa aaleaacaaIXaGaaG4maaqabaaaaa@3666@ comprend les poids globaux { w 3 k * } , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiWaaeaaca WG3bWaa0baaSqaaiaaiodacaWGRbaabaGaaiOkaaaaaOGaay5Eaiaa w2haaiaacYcaaaa@3A69@ il est peu probable qu’il ajoute une valeur importante à la variance totale en raison de la corrélation généralement faible entre les restes des phases 1 et 3. Par ailleurs, le terme C 23 , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4qamaaBa aaleaacaaIYaGaaG4maaqabaGccaGGSaaaaa@3721@ même s’il est pondéré par les poids globaux de 2 e MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOmamaaCa aaleqabaGaaeyzaaaaaaa@35CB@ phase seulement, peut être significatif en raison de la forte corrélation entre les restes des phases 2 et 3, car ils contiennent tous deux le terme x 3 k γ ^ 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaDa aaleaacaaIZaGaam4AaaqaaOGamai2gkdiIcaacuaHZoWzgaqcamaa BaaaleaacaaIZaaabeaaaaa@3C5B@ pour k s 3 . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4AaiabgI GiolaadohadaWgaaWcbaGaaG4maaqabaGccaGGUaaaaa@390B@ L’importance relative des termes pour certains plans généraux est spécifiée dans le tableau 3.1. Les coefficients γ , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4SdCMaai ilaaaa@3651@ qui englobent l’effet total des variables x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCiEaaaa@34FB@ qu’ils multiplient, prennent maintenant une forme plus intéressante et compliquée. Par exemple, γ ^ 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafq4SdCMbaK aadaWgaaWcbaGaaGymaaqabaaaaa@3698@ tient compte des projections de x 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCiEamaaBa aaleaacaaIXaaabeaaaaa@35E2@ sur x 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCiEamaaBa aaleaacaaIYaaabeaaaaa@35E3@ et de x 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCiEamaaBa aaleaacaaIXaaabeaaaaa@35E2@ sur x 3 , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCiEamaaBa aaleaacaaIZaaabeaakiaacYcaaaa@369E@ mais avec déduction de la projection de x 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCiEamaaBa aaleaacaaIXaaabeaaaaa@35E2@ sur la projection de x 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCiEamaaBa aaleaacaaIYaaabeaaaaa@35E3@ sur x 3 . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCiEamaaBa aaleaacaaIZaaabeaakiaac6caaaa@36A0@

Tableau 3.1
Une représentation générale de l’importance relative de chacun des termes dans (3.8) pour certains scénarios. Les points noirs indiquent une forte dominance, les points gris foncé, une dominance modérée et les points gris clair, une non-dominance
Sommaire du tableau
Le tableau montre les résultats de Une représentation générale de l’importance relative de chacun des termes dans (3.8) pour certains scénarios. Les points noirs indiquent une forte dominance. Les données sont présentées selon Cas (titres de rangée) et Description et (figurant comme en-tête de colonne).
Cas Description V1 V2 V3 C12 C13 C23
1 Pratiquement aucun échantillonnage supplémentaire aux deuxième et troisième phases : w 2 w 3 1. MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4DamaaBa aaleaacaaIYaaabeaakiabgIKi7kaadEhadaWgaaWcbaGaaG4maaqa baGccqGHijYUcaaIXaGaaiOlaaaa@3EE9@ Ceci est un cercle gris foncé Ceci est un cercle gris foncé Ceci est un cercle gris foncé Ceci est un cercle gris pâle Ceci est un cercle gris foncé Ceci est un cercle gris pâle
2 Les poids w 1 , w 2 , w 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4DamaaBa aaleaacaaIXaaabeaakiaaygW7caaISaGaaGPaVlaadEhadaWgaaWc baGaaGOmaaqabaGccaaMb8UaaGilaiaaykW7caWG3bWaaSbaaSqaai aaiodaaeqaaaaa@4392@ sont de taille modérée. Ceci est un cercle gris pâle Ceci est un cercle gris moyen Ceci est un cercle gris foncé Ceci est un cercle gris pâle Ceci est un cercle gris foncé Ceci est un cercle gris pâle
3 n 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBamaaBa aaleaacaaIZaaabeaaaaa@3818@ nettement plus petit que n 2 , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBamaaBa aaleaacaaIYaaabeaakiaacYcaaaa@38D1@ indépendamment des tailles de w 1 , w 2 . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4DamaaBa aaleaacaaIXaaabeaakiaaygW7caaISaGaaGPaVlaadEhadaWgaaWc baGaaGOmaaqabaGccaaMb8UaaiOlaaaa@401E@ Ceci est un cercle gris pâle Ceci est un cercle gris pâle Ceci est un cercle gris foncé Ceci est un cercle gris pâle Ceci est un cercle gris pâle Ceci est un cercle gris pâle

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