Tests pour évaluer le biais de non-réponse dans les enquêtes Section 2. Poststratification

2.1 Paramètre et variance par linéarisation

Supposons que la population finie U MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyvaaaa@3548@ contient H MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamisaaaa@353B@ strates, avec N h MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOtamaaBa aaleaacaWGObaabeaaaaa@365A@ unités primaires d’échantillonnage (UPE) dans la strate h , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiAaiaacY caaaa@360B@ M h i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamytamaaBa aaleaacaWGObGaamyAaaqabaaaaa@3747@ unités dans l’UPE i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAaaaa@355C@ de la strate h , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiAaiaacY caaaa@360B@ et M = h i M h i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamytaiaai2 dadaaeqaqabSqaaiaadIgacaWGPbaabeqdcqGHris5aOGaamytamaa BaaaleaacaWGObGaamyAaaqabaaaaa@3CA9@ unités au total. Soit y h i k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEamaaBa aaleaacaWGObGaamyAaiaadUgaaeqaaaaa@3863@ la variable d’intérêt pour l’unité k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4Aaaaa@355E@ dans l’UPE ( h i ) . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaaca WGObGaamyAaaGaayjkaiaawMcaaiaac6caaaa@3884@ Un échantillon probabiliste S MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uaaaa@3546@ est tiré de la population, avec n h MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBamaaBa aaleaacaWGObaabeaaaaa@367A@ UPE sélectionnées dans la strate h MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiAaaaa@355B@ et n = h = 1 H n h . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaiaai2 dadaaeWaqaaiaad6gadaWgaaWcbaGaamiAaaqabaaabaGaamiAaiaa i2dacaaIXaaabaGaamisaaqdcqGHris5aOGaaiOlaaaa@3E23@ L’échantillon d’UPE provenant de la strate h MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiAaaaa@355B@ est désigné par S h , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uamaaBa aaleaacaWGObaabeaakiaacYcaaaa@3719@ et l’échantillon d’unités provenant de l’UPE ( h i ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaaca WGObGaamyAaaGaayjkaiaawMcaaaaa@37D2@ est désigné par S h i . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uamaaBa aaleaacaWGObGaamyAaaqabaGccaGGUaaaaa@3809@ Chaque unité possède un poids de sondage w h i k = 1 / P ( unité h i k S ) , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9pf0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4DamaaBa aaleaacaWGObGaamyAaiaadUgaaeqaaOGaaGypamaalyaabaGaaGym aaqaaiaadcfaaaWaaeWaaeaacaqG1bGaaeOBaiaabMgacaqG0bGaae y6aiaaysW7caaMc8UaamiAaiaadMgacaWGRbGaeyicI4Saam4uaaGa ayjkaiaawMcaaiaacYcaaaa@4A78@ et le poids de sondage au niveau de l’UPE est w h i = 1 / P ( UPE h i S h ) . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9pf0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4DamaaBa aaleaacaWGObGaamyAaaqabaGccaaI9aWaaSGbaeaacaaIXaaabaGa amiuaaaadaqadaqaaiaabwfacaqGqbGaaeyraiaaysW7caaMc8Uaam iAaiaadMgacqGHiiIZcaWGtbWaaSbaaSqaaiaadIgaaeqaaaGccaGL OaGaayzkaaGaaiOlaaaa@46F8@

Deux cadres de référence sont utilisés fréquemment pour le mécanisme de non-réponse. Dans un cadre « en marche avant » à deux phases, l’échantillon est sélectionné à la phase 1 et le mécanisme de non-réponse représente une deuxième phase de sélection (Oh et Scheuren 1987; Särndal et Lundström 2005). Fay (1991) a proposé un cadre « en marche arrière » ou « cadre inversé» qui a été étudié plus en profondeur par Shao et Steel (1999) et Haziza, Thompson et Yung (2010). Dans ce cadre, le mécanisme de non-réponse est appliqué à la population finie pour commencer, puis l’échantillon est sélectionné. Le cadre inversé, que nous suivons dans le présent article, spécifie un mécanisme de non-réponse pour les unités non échantillonnées ainsi que les unités échantillonnées. Nous supposons que chaque unité de la population possède une valeur de l’indicateur de réponse r h i k . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCamaaBa aaleaacaWGObGaamyAaiaadUgaaeqaaOGaaiOlaaaa@3918@ Soit R h i k = E [ r h i k ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOuamaaBa aaleaacaWGObGaamyAaiaadUgaaeqaaOGaaGypaiaadweadaWadaqa aiaadkhadaWgaaWcbaGaamiAaiaadMgacaWGRbaabeaaaOGaay5wai aaw2faaaaa@3FC1@ sous le mécanisme de réponse dans la population finie, de sorte que R h i k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOuamaaBa aaleaacaWGObGaamyAaiaadUgaaeqaaaaa@383C@ est la valeur de la vraie propension à répondre de l’unité ( h i k ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaaca WGObGaamyAaiaadUgaaiaawIcacaGLPaaaaaa@38C2@ dans la population.

Supposons que la caractéristique y MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEaaaa@356C@ est connue pour toutes les unités dans l’échantillon sélectionné. Nous comparons le total de population estimé en utilisant chacune des unités dans l’échantillon au total estimé en utilisant les répondants pondérés par poststratification. Il existe C MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4qaaaa@3536@ poststrates, et la poststrate c MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4yaaaa@3556@ contient M c MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamytamaaBa aaleaacaWGJbaabeaaaaa@3654@ unités de la population avec M = c = 1 C M c . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamytaiaai2 dadaaeWaqaaiaad2eadaWgaaWcbaGaam4yaaqabaaabaGaam4yaiaa i2dacaaIXaaabaGaam4qaaqdcqGHris5aOGaaiOlaaaa@3DD2@ Les dénombrements de poststrate M c MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamytamaaBa aaleaacaWGJbaabeaaaaa@3654@ peuvent être obtenus d’après les données de la base de sondage si les variables de poststratification sont connues pour chaque unité figurant dans la base. Souvent, cependant, les dénombrements de poststrate proviennent d’une source extérieure, telle qu’un recensement. Soit δ c h i k = 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiTdq2aaS baaSqaaiaadogacaWGObGaamyAaiaadUgaaeqaaOGaaGypaiaaigda aaa@3B7E@ si l’unité ( h i k ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaaca WGObGaamyAaiaadUgaaiaawIcacaGLPaaaaaa@38C2@ se trouve dans la poststrate c MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4yaaaa@3556@ et 0 autrement. Le taux de réponse de la population dans la poststrate c MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4yaaaa@3556@ est p c = h i k U R h i k δ c h i k / M c . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCamaaBa aaleaacaWGJbaabeaakiaai2dadaWcgaqaamaaqababaGaamOuamaa BaaaleaacaWGObGaamyAaiaadUgaaeqaaOGaeqiTdq2aaSbaaSqaai aadogacaWGObGaamyAaiaadUgaaeqaaaqaaiaadIgacaWGPbGaam4A aiabgIGiolaadwfaaeqaniabggHiLdaakeaacaWGnbWaaSbaaSqaai aadogaaeqaaaaakiaac6caaaa@4A67@ Yung et Rao (2000) ont supposé que le taux de réponse p c MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCamaaBa aaleaacaWGJbaabeaaaaa@3677@ était le même pour chaque poststrate. Dans de nombreuses applications, néanmoins, les poststrates sont formées de manière que les propensions à répondre soient homogènes à l’intérieur de chaque poststrate, mais que les poststrates proprement dites possèdent des propensions à répondre moyennes différentes. Par conséquent, nous permettons à p c MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCamaaBa aaleaacaWGJbaabeaaaaa@3677@ de différer d’une poststrate à l’autre.

Si y MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEaaaa@356C@ est connue pour tous les membres de l’échantillon sélectionné, l’estimateur du total de population en utilisant l’échantillon est donné par

Y ^ S S = h i k S w h i k y h i k = h i k U Z h i k w h i k y h i k , ( 2.1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmywayaaja WaaSbaaSqaaiaadofacaWGtbaabeaakiaai2dadaaeqbqaaiaadEha daWgaaWcbaGaamiAaiaadMgacaWGRbaabeaakiaadMhadaWgaaWcba GaamiAaiaadMgacaWGRbaabeaaaeaacaWGObGaamyAaiaadUgacqGH iiIZcaWGtbaabeqdcqGHris5aOGaaGypamaaqafabaGaamOwamaaBa aaleaacaWGObGaamyAaiaadUgaaeqaaOGaam4DamaaBaaaleaacaWG ObGaamyAaiaadUgaaeqaaOGaamyEamaaBaaaleaacaWGObGaamyAai aadUgaaeqaaaqaaiaadIgacaWGPbGaam4AaiabgIGiolaadwfaaeqa niabggHiLdGccaaISaGaaGzbVlaaywW7caaMf8UaaGzbVlaaywW7ca GGOaGaaGOmaiaac6cacaaIXaGaaiykaaaa@6725@

w h i k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4DamaaBa aaleaacaWGObGaamyAaiaadUgaaeqaaaaa@3861@ est le poids de sondage de l’unité k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4Aaaaa@355E@ de l’UPE i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAaaaa@355C@ dans la strate h MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiAaaaa@355B@ et Z h i k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOwamaaBa aaleaacaWGObGaamyAaiaadUgaaeqaaaaa@3844@ est la variable indicatrice d’inclusion dans l’échantillon. En utilisant uniquement les répondants, l’estimateur poststratifié du total de population est donné par

Y ^ P S = c = 1 C M c h i k S w h i k r h i k δ c h i k y h i k h i k S w h i k r h i k δ c h i k = c = 1 C M c Y ^ c R M ^ c R . ( 2.2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmywayaaja WaaSbaaSqaaiaadcfacaWGtbaabeaakiaai2dadaaeWbqaaiaad2ea daWgaaWcbaGaam4yaaqabaaabaGaam4yaiaai2dacaaIXaaabaGaam 4qaaqdcqGHris5aOWaaSaaaeaadaaeqbqaaiaadEhadaWgaaWcbaGa amiAaiaadMgacaWGRbaabeaakiaadkhadaWgaaWcbaGaamiAaiaadM gacaWGRbaabeaakiabes7aKnaaBaaaleaacaWGJbGaamiAaiaadMga caWGRbaabeaakiaadMhadaWgaaWcbaGaamiAaiaadMgacaWGRbaabe aaaeaacaWGObGaamyAaiaadUgacqGHiiIZcaWGtbaabeqdcqGHris5 aaGcbaWaaabuaeaacaWG3bWaaSbaaSqaaiaadIgacaWGPbGaam4Aaa qabaGccaWGYbWaaSbaaSqaaiaadIgacaWGPbGaam4AaaqabaGccqaH 0oazdaWgaaWcbaGaam4yaiaadIgacaWGPbGaam4AaaqabaaabaGaam iAaiaadMgacaWGRbGaeyicI4Saam4uaaqab0GaeyyeIuoaaaGccaaI 9aWaaabCaeaacaWGnbWaaSbaaSqaaiaadogaaeqaaaqaaiaadogaca aI9aGaaGymaaqaaiaadoeaa0GaeyyeIuoakmaalaaabaGabmywayaa jaWaa0baaSqaaiaadogaaeaacaWGsbaaaaGcbaGabmytayaajaWaa0 baaSqaaiaadogaaeaacaWGsbaaaaaakiaai6cacaaMf8UaaGzbVlaa ywW7caaMf8UaaGzbVlaacIcacaaIYaGaaiOlaiaaikdacaGGPaaaaa@86DE@

Nous définissons le paramètre de population finie d’intérêt comme étant la différence entre la valeur espérée de Y ^ P S MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmywayaaja WaaSbaaSqaaiaadcfacaWGtbaabeaaaaa@3735@ et la valeur espérée de Y ^ S S , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmywayaaja WaaSbaaSqaaiaadofacaWGtbaabeaakiaacYcaaaa@37F2@ qui sera 0 s’il n’existe aucun biais de non-réponse après la poststratification. Définissons

M c R = h i k U δ c h i k R h i k = p c M c , Y c R = h i k U δ c h i k R h i k y h i k , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqbaeaabiGaaa qaaiaad2eadaqhaaWcbaGaam4yaaqaaiaadkfaaaaakeaacaaI9aWa aabuaeaacqaH0oazdaWgaaWcbaGaam4yaiaadIgacaWGPbGaam4Aaa qabaGccaWGsbWaaSbaaSqaaiaadIgacaWGPbGaam4AaaqabaaabaGa amiAaiaadMgacaWGRbGaeyicI4Saamyvaaqab0GaeyyeIuoakiaai2 dacaWGWbWaaSbaaSqaaiaadogaaeqaaOGaamytamaaBaaaleaacaWG JbaabeaakiaaiYcaaeaacaWGzbWaa0baaSqaaiaadogaaeaacaWGsb aaaaGcbaGaaGypamaaqafabaGaeqiTdq2aaSbaaSqaaiaadogacaWG ObGaamyAaiaadUgaaeqaaOGaamOuamaaBaaaleaacaWGObGaamyAai aadUgaaeqaaOGaamyEamaaBaaaleaacaWGObGaamyAaiaadUgaaeqa aaqaaiaadIgacaWGPbGaam4AaiabgIGiolaadwfaaeqaniabggHiLd GccaaISaaaaaaa@66D7@

et

θ = c = 1 C M c Y c R M c R Y = c = 1 C Y c R p c Y . ( 2.3 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiUdeNaaG ypamaaqahabaGaamytamaaBaaaleaacaWGJbaabeaaaeaacaWGJbGa aGypaiaaigdaaeaacaWGdbaaniabggHiLdGcdaWcaaqaaiaadMfada qhaaWcbaGaam4yaaqaaiaadkfaaaaakeaacaWGnbWaa0baaSqaaiaa dogaaeaacaWGsbaaaaaakiabgkHiTiaadMfacaaI9aWaaabCaeaada WcaaqaaiaadMfadaqhaaWcbaGaam4yaaqaaiaadkfaaaaakeaacaWG WbWaaSbaaSqaaiaadogaaeqaaaaaaeaacaWGJbGaaGypaiaaigdaae aacaWGdbaaniabggHiLdGccqGHsislcaWGzbGaaGOlaiaaywW7caaM f8UaaGzbVlaaywW7caaMf8UaaiikaiaaikdacaGGUaGaaG4maiaacM caaaa@5EAE@

En utilisant la relation h i k U δ c h i k ( R h i k p c ) = 0, MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaabeaeaacq aH0oazdaWgaaWcbaGaam4yaiaadIgacaWGPbGaam4AaaqabaaabaGa amiAaiaadMgacaWGRbGaeyicI4Saamyvaaqab0GaeyyeIuoakmaabm aabaGaamOuamaaBaaaleaacaWGObGaamyAaiaadUgaaeqaaOGaeyOe I0IaamiCamaaBaaaleaacaWGJbaabeaaaOGaayjkaiaawMcaaiaai2 dacaaIWaGaaGilaaaa@4B95@

θ = c = 1 C h i k U y h i k δ c h i k ( R h i k p c 1 ) = c = 1 C h i k U δ c h i k ( R h i k p c 1 ) ( y h i k Y c R M c R ) . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqbaeaabiGaaa qaaiabeI7aXbqaaiaai2dadaaeWbqabSqaaiaadogacaaI9aGaaGym aaqaaiaadoeaa0GaeyyeIuoakmaaqafabaGaamyEamaaBaaaleaaca WGObGaamyAaiaadUgaaeqaaOGaeqiTdq2aaSbaaSqaaiaadogacaWG ObGaamyAaiaadUgaaeqaaaqaaiaadIgacaWGPbGaam4AaiabgIGiol aadwfaaeqaniabggHiLdGcdaqadaqaamaalaaabaGaamOuamaaBaaa leaacaWGObGaamyAaiaadUgaaeqaaaGcbaGaamiCamaaBaaaleaaca WGJbaabeaaaaGccqGHsislcaaIXaaacaGLOaGaayzkaaaabaaabaGa aGypamaaqahabeWcbaGaam4yaiaai2dacaaIXaaabaGaam4qaaqdcq GHris5aOWaaabuaeaacqaH0oazdaWgaaWcbaGaam4yaiaadIgacaWG PbGaam4AaaqabaaabaGaamiAaiaadMgacaWGRbGaeyicI4Saamyvaa qab0GaeyyeIuoakmaabmaabaWaaSaaaeaacaWGsbWaaSbaaSqaaiaa dIgacaWGPbGaam4AaaqabaaakeaacaWGWbWaaSbaaSqaaiaadogaae qaaaaakiabgkHiTiaaigdaaiaawIcacaGLPaaadaqadaqaaiaadMha daWgaaWcbaGaamiAaiaadMgacaWGRbaabeaakiabgkHiTmaalaaaba GaamywamaaDaaaleaacaWGJbaabaGaamOuaaaaaOqaaiaad2eadaqh aaWcbaGaam4yaaqaaiaadkfaaaaaaaGccaGLOaGaayzkaaGaaGOlaa aaaaa@7F8C@

Nous cherchons à tester l’hypothèse H 0 : θ = 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamisamaaBa aaleaacaaIWaaabeaakiaaykW7caaI6aGaeqiUdeNaaGypaiaaicda aaa@3BB1@ c. H A : θ 0 , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamisamaaBa aaleaacaWGbbaabeaakiaaykW7caaI6aGaeqiUdeNaeyiyIKRaaGim aiaacYcaaaa@3D6D@ ou alternativement à obtenir un intervalle de confiance pour θ . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiUdeNaai Olaaaa@36D6@ Si la propension à répondre dans chaque poststrate c MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4yaaaa@3556@ est uniforme avec R h i k = p c MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOuamaaBa aaleaacaWGObGaamyAaiaadUgaaeqaaOGaaGypaiaadchadaWgaaWc baGaam4yaaqabaaaaa@3B16@ pour toutes les unités ayant δ c h i k = 1 , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiTdq2aaS baaSqaaiaadogacaWGObGaamyAaiaadUgaaeqaaOGaaGypaiaaigda caGGSaaaaa@3C2E@ alors la différence θ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiUdehaaa@3624@ sera nulle. Alternativement, θ = 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiUdeNaaG ypaiaaicdaaaa@37A5@ s’il n’existe aucune variabilité dans la variable de réponse y h i k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEamaaBa aaleaacaWGObGaamyAaiaadUgaaeqaaaaa@3863@ dans chaque poststrate. Si l’une ou l’autre de ces conditions est vérifiée, la poststratification corrige le biais de non-réponse. Notons que, si la propension à répondre est uniforme dans chacune des poststrates  MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGWj0Jf9crFfpeea0xh9v8qiW7rqqrpu0xh9Wqpm0db9Wq pepeuf0xe9q8qiYRWFGCk9vi=dbvc9s8vr0db9Fn0dbbG8Fq0Jfr=x fr=xfbpdbiqaaeaaciGaaiaabeqaamaabaabaaGcbaacbaqcLbwaqa aaaaaaaaWdbiaa=nbiaaa@3D01@ c’est-à-dire si les variables de poststratification expliquent complètement la variabilité des propensions à répondre sous-jacentes  MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGWj0Jf9crFfpeea0xh9v8qiW7rqqrpu0xh9Wqpm0db9Wq pepeuf0xe9q8qiYRWFGCk9vi=dbvc9s8vr0db9Fn0dbbG8Fq0Jfr=x fr=xfbpdbiqaaeaaciGaaiaabeqaamaabaabaaGcbaacbaqcLbwaqa aaaaaaaaWdbiaa=nbiaaa@3D01@ , alors la poststratification éliminera effectivement le biais pour chaque variable y MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEaaaa@356C@ possible. Si la variance de y h i k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEamaaBa aaleaacaWGObGaamyAaiaadUgaaeqaaaaa@3863@ est 0 dans chaque poststrate, la poststratification élimine le biais pour y , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEaiaacY caaaa@361C@ mais n’élimine pas nécessairement le biais pour les autres variables.

Nous estimons θ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiUdehaaa@3624@ par θ ^ = Y ^ P S Y ^ S S , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiUdeNbaK aacaaI9aGabmywayaajaWaaSbaaSqaaiaadcfacaWGtbaabeaakiab gkHiTiqadMfagaqcamaaBaaaleaacaWGtbGaam4uaaqabaGccaGGSa aaaa@3E3D@ qui peut se réécrire sous la forme

θ ^ = Y ^ P S Y ^ S S = c = 1 C 1 p c ( Y ^ c R Y ¯ c R ( M ^ c R M c R ) + T ^ c ) Y ^ S S , ( 2.4 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiUdeNbaK aacaaI9aGabmywayaajaWaaSbaaSqaaiaadcfacaWGtbaabeaakiab gkHiTiqadMfagaqcamaaBaaaleaacaWGtbGaam4uaaqabaGccaaI9a WaaabCaeqaleaacaWGJbGaaGypaiaaigdaaeaacaWGdbaaniabggHi LdGcdaWcaaqaaiaaigdaaeaacaWGWbWaaSbaaSqaaiaadogaaeqaaa aakmaabmaabaGabmywayaajaWaa0baaSqaaiaadogaaeaacaWGsbaa aOGaeyOeI0IabmywayaaraWaa0baaSqaaiaadogaaeaacaWGsbaaaO WaaeWaaeaaceWGnbGbaKaadaqhaaWcbaGaam4yaaqaaiaadkfaaaGc cqGHsislcaWGnbWaa0baaSqaaiaadogaaeaacaWGsbaaaaGccaGLOa GaayzkaaGaey4kaSIabmivayaajaWaaSbaaSqaaiaadogaaeqaaaGc caGLOaGaayzkaaGaeyOeI0IabmywayaajaWaaSbaaSqaaiaadofaca WGtbaabeaakiaaiYcacaaMf8UaaGzbVlaaywW7caaMf8UaaGzbVlaa cIcacaaIYaGaaiOlaiaaisdacaGGPaaaaa@69B7@

Y ¯ c R = Y c R / M c R , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmywayaara Waa0baaSqaaiaadogaaeaacaWGsbaaaOGaaGypamaalyaabaGaamyw amaaDaaaleaacaWGJbaabaGaamOuaaaaaOqaaiaad2eadaqhaaWcba Gaam4yaaqaaiaadkfaaaaaaOGaaiilaaaa@3E83@ y ¯ c R = Y ^ c R / M ^ c R , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmyEayaara Waa0baaSqaaiaadogaaeaacaWGsbaaaOGaaGypamaalyaabaGabmyw ayaajaWaa0baaSqaaiaadogaaeaacaWGsbaaaaGcbaGabmytayaaja Waa0baaSqaaiaadogaaeaacaWGsbaaaaaakiaacYcaaaa@3EC3@ et

T ^ c = ( y ¯ c R Y ¯ c R ) ( M ^ c R M c R ) . ( 2.5 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmivayaaja WaaSbaaSqaaiaadogaaeqaaOGaaGypaiabgkHiTmaabmaabaGabmyE ayaaraWaa0baaSqaaiaadogaaeaacaWGsbaaaOGaeyOeI0Iabmyway aaraWaa0baaSqaaiaadogaaeaacaWGsbaaaaGccaGLOaGaayzkaaWa aeWaaeaaceWGnbGbaKaadaqhaaWcbaGaam4yaaqaaiaadkfaaaGccq GHsislcaWGnbWaa0baaSqaaiaadogaaeaacaWGsbaaaaGccaGLOaGa ayzkaaGaaGOlaiaaywW7caaMf8UaaGzbVlaaywW7caaMf8Uaaiikai aaikdacaGGUaGaaGynaiaacMcaaaa@54B0@

Le théorème 1 donne la variance de θ ^ . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiUdeNbaK aacaGGUaaaaa@36E6@ Définissons

e R h i k = c = 1 C δ c h i k { R h i k p c ( y h i k Y ¯ c R ) y h i k } . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyzamaaBa aaleaacaWGsbGaamiAaiaadMgacaWGRbaabeaakiaai2dadaaeWbqa aiabes7aKnaaBaaaleaacaWGJbGaamiAaiaadMgacaWGRbaabeaaae aacaWGJbGaaGypaiaaigdaaeaacaWGdbaaniabggHiLdGcdaGadaqa amaalaaabaGaamOuamaaBaaaleaacaWGObGaamyAaiaadUgaaeqaaa GcbaGaamiCamaaBaaaleaacaWGJbaabeaaaaGcdaqadaqaaiaadMha daWgaaWcbaGaamiAaiaadMgacaWGRbaabeaakiabgkHiTiqadMfaga qeamaaDaaaleaacaWGJbaabaGaamOuaaaaaOGaayjkaiaawMcaaiab gkHiTiaadMhadaWgaaWcbaGaamiAaiaadMgacaWGRbaabeaaaOGaay 5Eaiaaw2haaiaai6caaaa@5C1E@

Nous supposons les conditions de régularité suivantes.

Les hypothèses (A1) et (A4) font en sorte que le dénominateur dans (2.3) soit presque certainement non nul. L’hypothèse (A2) pourrait être remplacée par des conditions de type Liapunov plus faibles, telles que celles énoncées dans le théorème 1.3.2 de Fuller (2009) ou dans Yung et Rao (2000) si des hypothèses plus contraignantes sont appliquées à la structure de covariance des indicateurs de réponse r h i k ; MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCamaaBa aaleaacaWGObGaamyAaiaadUgaaeqaaOGaai4oaaaa@3925@ cependant, en pratique, on peut supposer que presque toute caractéristique mesurée dans une population finie est bornée. L’hypothèse (A5) est plus faible que l’hypothèse utilisée dans Kim et Kim (2007) voulant que les indicateurs de réponse soient indépendants entre les unités. Sous l’hypothèse (A5), les individus se trouvant dans la même UPE (par exemple, des personnes vivant dans le même ménage ou dans la même ville) peuvent présenter une dépendance lorsqu’elles choisissent de répondre ou non à l’enquête, mais les indicateurs de réponse des individus dans différentes UPE sont indépendants.

Théorème 1. Sous les conditions (A1) à (A5), la variance de θ ^ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiUdeNbaK aaaaa@3634@  est

V ( θ ^ ) = V 1 ( θ ^ ) + V 2 ( θ ^ ) , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOvamaabm aabaGafqiUdeNbaKaaaiaawIcacaGLPaaacaaI9aGaamOvamaaBaaa leaacaaIXaaabeaakmaabmaabaGafqiUdeNbaKaaaiaawIcacaGLPa aacqGHRaWkcaWGwbWaaSbaaSqaaiaaikdaaeqaaOWaaeWaaeaacuaH 4oqCgaqcaaGaayjkaiaawMcaaiaaiYcaaaa@452D@

V 1 ( θ ^ ) = V ( h i k U Z h i k w h i k e R h i k ) + E [ V [ h i k U Z h i k w h i k c = 1 C δ c h i k r h i k p c ( y h i k Y ¯ c R ) | Z ] ] ( 2.6 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOvamaaBa aaleaacaaIXaaabeaakmaabmaabaGafqiUdeNbaKaaaiaawIcacaGL PaaacaaI9aGaamOvamaabmaabaWaaabuaeaacaWGAbWaaSbaaSqaai aadIgacaWGPbGaam4AaaqabaGccaWG3bWaaSbaaSqaaiaadIgacaWG PbGaam4AaaqabaGccaWGLbWaaSbaaSqaaiaadkfacaWGObGaamyAai aadUgaaeqaaaqaaiaadIgacaWGPbGaam4AaiabgIGiolaadwfaaeqa niabggHiLdaakiaawIcacaGLPaaacqGHRaWkcaWGfbWaamWaaeaaca WGwbWaamWaaeaadaabcaqaamaaqafabaGaamOwamaaBaaaleaacaWG ObGaamyAaiaadUgaaeqaaOGaam4DamaaBaaaleaacaWGObGaamyAai aadUgaaeqaaaqaaiaadIgacaWGPbGaam4AaiabgIGiolaadwfaaeqa niabggHiLdGcdaaeWbqaaiabes7aKnaaBaaaleaacaWGJbGaamiAai aadMgacaWGRbaabeaaaeaacaWGJbGaaGypaiaaigdaaeaacaWGdbaa niabggHiLdGcdaWcaaqaaiaadkhadaWgaaWcbaGaamiAaiaadMgaca WGRbaabeaaaOqaaiaadchadaWgaaWcbaGaam4yaaqabaaaaOWaaeWa aeaacaWG5bWaaSbaaSqaaiaadIgacaWGPbGaam4AaaqabaGccqGHsi slceWGzbGbaebadaqhaaWcbaGaam4yaaqaaiaadkfaaaaakiaawIca caGLPaaacaaMc8oacaGLiWoacaaMc8UaaCOwaaGaay5waiaaw2faaa Gaay5waiaaw2faaiaaywW7caaMf8UaaGzbVlaaywW7caaMf8Uaaiik aiaaikdacaGGUaGaaGOnaiaacMcaaaa@916B@

et

V 2 ( θ ^ ) = V [ c = 1 C T ^ c p c ] + 2 Cov [ c = 1 C T ^ c p c , c = 1 C ( y ¯ c R Y ¯ c R ) M ^ c R p c Y ^ S S ] = o ( M 2 / n ) . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOvamaaBa aaleaacaaIYaaabeaakmaabmaabaGafqiUdeNbaKaaaiaawIcacaGL PaaacaaI9aGaamOvamaadmaabaWaaabCaeqaleaacaWGJbGaaGypai aaigdaaeaacaWGdbaaniabggHiLdGcdaWcaaqaaiqadsfagaqcamaa BaaaleaacaWGJbaabeaaaOqaaiaadchadaWgaaWcbaGaam4yaaqaba aaaaGccaGLBbGaayzxaaGaey4kaSIaaGOmaiaaysW7caaMc8Uaae4q aiaab+gacaqG2bWaamWaaeaadaaeWbqabSqaaiaadogacaaI9aGaaG ymaaqaaiaadoeaa0GaeyyeIuoakmaalaaabaGabmivayaajaWaaSba aSqaaiaadogaaeqaaaGcbaGaamiCamaaBaaaleaacaWGJbaabeaaaa GccaaISaWaaabCaeqaleaacaWGJbGaaGypaiaaigdaaeaacaWGdbaa niabggHiLdGcdaWcaaqaamaabmaabaGabmyEayaaraWaa0baaSqaai aadogaaeaacaWGsbaaaOGaeyOeI0IabmywayaaraWaa0baaSqaaiaa dogaaeaacaWGsbaaaaGccaGLOaGaayzkaaGabmytayaajaWaa0baaS qaaiaadogaaeaacaWGsbaaaaGcbaGaamiCamaaBaaaleaacaWGJbaa beaaaaGccqGHsislceWGzbGbaKaadaWgaaWcbaGaam4uaiaadofaae qaaaGccaGLBbGaayzxaaGaaGypaiaad+gadaqadaqaamaalyaabaGa amytamaaCaaaleqabaGaaGOmaaaaaOqaaiaad6gaaaaacaGLOaGaay zkaaGaaGOlaaaa@77F1@

La preuve est donnée en annexe. Habituellement, seule V 1 ( θ ^ ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOvamaaBa aaleaacaaIXaaabeaakmaabmaabaGafqiUdeNbaKaaaiaawIcacaGL Paaaaaa@3989@ serait prise en considération, parce que pour la plupart des applications, elle est d’un ordre plus élevé que V 2 ( θ ^ ) . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOvamaaBa aaleaacaaIYaaabeaakmaabmaabaGafqiUdeNbaKaaaiaawIcacaGL PaaacaGGUaaaaa@3A3C@ Par contre, contrairement aux situations étudiées habituellement dans les sondages, le terme d’ordre un de la variance par linéarisation peut être nul dans certains cas pour lesquels on a alors V ( θ ^ ) = V 2 ( θ ^ ) . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOvamaabm aabaGafqiUdeNbaKaaaiaawIcacaGLPaaacaaI9aGaamOvamaaBaaa leaacaaIYaaabeaakmaabmaabaGafqiUdeNbaKaaaiaawIcacaGLPa aacaGGUaaaaa@3F2D@ Si le terme d’ordre un n’est pas exactement nul, mais est o ( M 2 / n ) , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4Bamaabm aabaWaaSGbaeaacaWGnbWaaWbaaSqabeaacaaIYaaaaaGcbaGaamOB aaaaaiaawIcacaGLPaaacaGGSaaaaa@3A69@ les deux termes de variance sont nécessaires.

Dans (2.6), le deuxième terme est égal à 0 si p c = 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCamaaBa aaleaacaWGJbaabeaakiaai2dacaaIXaaaaa@3803@ pour toutes les poststrates c MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4yaaaa@3556@ (c’est-à-dire que la réponse est complète), ou si les valeurs de y MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEaaaa@356C@ dans la poststrate c MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4yaaaa@3556@ ne présentent aucune variabilité pour chaque poststrate pour laquelle p c < 1. MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCamaaBa aaleaacaWGJbaabeaakiaaiYdacaaIXaGaaiOlaaaa@38B4@ Si les indicateurs de réponse r h i k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCamaaBa aaleaacaWGObGaamyAaiaadUgaaeqaaaaa@385C@ sont tous indépendants, alors

E [ V ( h i k U Z h i k w h i k c = 1 C δ c h i k r h i k p c ( y h i k Y ¯ c R ) | Z ) ] = h i k U w h i k c = 1 C δ c h i k 1 p c p c ( y h i k Y ¯ c R ) 2 . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyramaadm aabaGaamOvamaabmaabaWaaqGaaeaadaaeqbqaaiaadQfadaWgaaWc baGaamiAaiaadMgacaWGRbaabeaakiaadEhadaWgaaWcbaGaamiAai aadMgacaWGRbaabeaaaeaacaWGObGaamyAaiaadUgacqGHiiIZcaWG vbaabeqdcqGHris5aOWaaabCaeaacqaH0oazdaWgaaWcbaGaam4yai aadIgacaWGPbGaam4AaaqabaaabaGaam4yaiaai2dacaaIXaaabaGa am4qaaqdcqGHris5aOWaaSaaaeaacaWGYbWaaSbaaSqaaiaadIgaca WGPbGaam4AaaqabaaakeaacaWGWbWaaSbaaSqaaiaadogaaeqaaaaa kmaabmaabaGaamyEamaaBaaaleaacaWGObGaamyAaiaadUgaaeqaaO GaeyOeI0IabmywayaaraWaa0baaSqaaiaadogaaeaacaWGsbaaaaGc caGLOaGaayzkaaGaaGPaVdGaayjcSdGaaGPaVlaahQfaaiaawIcaca GLPaaaaiaawUfacaGLDbaacaaI9aWaaabuaeaacaWG3bWaaSbaaSqa aiaadIgacaWGPbGaam4AaaqabaaabaGaamiAaiaadMgacaWGRbGaey icI4Saamyvaaqab0GaeyyeIuoakmaaqahabaGaeqiTdq2aaSbaaSqa aiaadogacaWGObGaamyAaiaadUgaaeqaaaqaaiaadogacaaI9aGaaG ymaaqaaiaadoeaa0GaeyyeIuoakmaalaaabaGaaGymaiabgkHiTiaa dchadaWgaaWcbaGaam4yaaqabaaakeaacaWGWbWaaSbaaSqaaiaado gaaeqaaaaakmaabmaabaGaamyEamaaBaaaleaacaWGObGaamyAaiaa dUgaaeqaaOGaeyOeI0IabmywayaaraWaa0baaSqaaiaadogaaeaaca WGsbaaaaGccaGLOaGaayzkaaWaaWbaaSqabeaacaaIYaaaaOGaaGOl aaaa@9093@

Sous le mécanisme de propension à répondre uniforme supposé voulant que R h i k = p c MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOuamaaBa aaleaacaWGObGaamyAaiaadUgaaeqaaOGaaGypaiaadchadaWgaaWc baGaam4yaaqabaaaaa@3B16@ pour toutes les unités de la population dans la poststrate c , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4yaiaacY caaaa@3606@ le premier terme dans (2.6) est

V ( h i k U Z h i k w h i k e R h i k ) = V { h i k U Z h i k w h i k c = 1 C δ c h i k ( Y ¯ c R ) } = V ( c = 1 C M ^ c Y ¯ c R ) . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOvamaabm aabaWaaabuaeaacaWGAbWaaSbaaSqaaiaadIgacaWGPbGaam4Aaaqa baGccaWG3bWaaSbaaSqaaiaadIgacaWGPbGaam4AaaqabaGccaWGLb WaaSbaaSqaaiaadkfacaWGObGaamyAaiaadUgaaeqaaaqaaiaadIga caWGPbGaam4AaiabgIGiolaadwfaaeqaniabggHiLdaakiaawIcaca GLPaaacaaI9aGaamOvamaacmaabaWaaabuaeaacaWGAbWaaSbaaSqa aiaadIgacaWGPbGaam4AaaqabaGccaWG3bWaaSbaaSqaaiaadIgaca WGPbGaam4AaaqabaaabaGaamiAaiaadMgacaWGRbGaeyicI4Saamyv aaqab0GaeyyeIuoakmaaqahabaGaeqiTdq2aaSbaaSqaaiaadogaca WGObGaamyAaiaadUgaaeqaaaqaaiaadogacaaI9aGaaGymaaqaaiaa doeaa0GaeyyeIuoakmaabmaabaGaeyOeI0IabmywayaaraWaa0baaS qaaiaadogaaeaacaWGsbaaaaGccaGLOaGaayzkaaaacaGL7bGaayzF aaGaaGypaiaadAfadaqadaqaamaaqahabaGabmytayaajaWaaSbaaS qaaiaadogaaeqaaOGabmywayaaraWaa0baaSqaaiaadogaaeaacaWG sbaaaaqaaiaadogacaaI9aGaaGymaaqaaiaadoeaa0GaeyyeIuoaaO GaayjkaiaawMcaaiaai6caaaa@7C33@

Si les propensions à répondre sont uniformes, ce terme est nul si la moyenne de population de Y ¯ c R MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmywayaara Waa0baaSqaaiaadogaaeaacaWGsbaaaaaa@3750@ est la même pour toutes les poststrates et que la somme des tailles estimées des poststrates est égale à M . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamytaiaac6 caaaa@35F2@

Si ( n / M 2 ) V 1 ( θ ^ ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaada Wcgaqaaiaad6gaaeaacaWGnbWaaWbaaSqabeaacaaIYaaaaaaaaOGa ayjkaiaawMcaaiaadAfadaWgaaWcbaGaaGymaaqabaGcdaqadaqaai qbeI7aXzaajaaacaGLOaGaayzkaaaaaa@3DE0@ converge vers une constante positive, un estimateur de variance par linéarisation pour V ( θ ^ ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOvamaabm aabaGafqiUdeNbaKaaaiaawIcacaGLPaaaaaa@3898@ est donné par

V ^ L ( θ ^ ) = h = 1 H n h n h 1 i S h ( b h i b h ) 2 ( 2.7 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOvayaaja WaaSbaaSqaaiaadYeaaeqaaOWaaeWaaeaacuaH4oqCgaqcaaGaayjk aiaawMcaaiaai2dadaaeWbqabSqaaiaadIgacaaI9aGaaGymaaqaai aadIeaa0GaeyyeIuoakmaalaaabaGaamOBamaaBaaaleaacaWGObaa beaaaOqaaiaad6gadaWgaaWcbaGaamiAaaqabaGccqGHsislcaaIXa aaamaaqafabeWcbaGaamyAaiabgIGiolaadofadaWgaaadbaGaamiA aaqabaaaleqaniabggHiLdGcdaqadaqaaiaadkgadaWgaaWcbaGaam iAaiaadMgaaeqaaOGaeyOeI0IaamOyamaaBaaaleaacaWGObaabeaa aOGaayjkaiaawMcaamaaCaaaleqabaGaaGOmaaaakiaaywW7caaMf8 UaaGzbVlaaywW7caaMf8UaaiikaiaaikdacaGGUaGaaG4naiaacMca aaa@6038@

b h i = k S h i w h i k { c = 1 C M c M ^ c R r h i k δ c h i k ( y h i k y ¯ c R ) y h i k } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOyamaaBa aaleaacaWGObGaamyAaaqabaGccaaI9aWaaabuaeaacaWG3bWaaSba aSqaaiaadIgacaWGPbGaam4AaaqabaaabaGaam4AaiabgIGiolaado fadaWgaaadbaGaamiAaiaadMgaaeqaaaWcbeqdcqGHris5aOWaaiWa aeaadaaeWbqabSqaaiaadogacaaI9aGaaGymaaqaaiaadoeaa0Gaey yeIuoakmaalaaabaGaamytamaaBaaaleaacaWGJbaabeaaaOqaaiqa d2eagaqcamaaDaaaleaacaWGJbaabaGaamOuaaaaaaGccaWGYbWaaS baaSqaaiaadIgacaWGPbGaam4AaaqabaGccqaH0oazdaWgaaWcbaGa am4yaiaadIgacaWGPbGaam4AaaqabaGcdaqadaqaaiaadMhadaWgaa WcbaGaamiAaiaadMgacaWGRbaabeaakiabgkHiTiqadMhagaqeamaa DaaaleaacaWGJbaabaGaamOuaaaaaOGaayjkaiaawMcaaiabgkHiTi aadMhadaWgaaWcbaGaamiAaiaadMgacaWGRbaabeaaaOGaay5Eaiaa w2haaaaa@681B@

et

b h = 1 n h i S h b h i . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOyamaaBa aaleaacaWGObaabeaakiaai2dadaWcaaqaaiaaigdaaeaacaWGUbWa aSbaaSqaaiaadIgaaeqaaaaakmaaqafabaGaamOyamaaBaaaleaaca WGObGaamyAaaqabaaabaGaamyAaiabgIGiolaadofadaWgaaadbaGa amiAaaqabaaaleqaniabggHiLdGccaaIUaaaaa@4456@

Théorème 2. Supposons que les conditions (A1) à (A5) sont vérifiées et que ( n / M 2 ) V 1 ( θ ^ ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaada Wcgaqaaiaad6gaaeaacaWGnbWaaWbaaSqabeaacaaIYaaaaaaaaOGa ayjkaiaawMcaaiaadAfadaWgaaWcbaGaaGymaaqabaGcdaqadaqaai qbeI7aXzaajaaacaGLOaGaayzkaaaaaa@3DE0@  converge vers une constante positive. Alors, ( n / M 2 ) [ V ^ L ( θ ^ ) V 1 ( θ ^ ) ] 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaada Wcgaqaaiaad6gaaeaacaWGnbWaaWbaaSqabeaacaaIYaaaaaaaaOGa ayjkaiaawMcaamaadmaabaGabmOvayaajaWaaSbaaSqaaiaadYeaae qaaOWaaeWaaeaacuaH4oqCgaqcaaGaayjkaiaawMcaaiabgkHiTiaa dAfadaWgaaWcbaGaaGymaaqabaGcdaqadaqaaiqbeI7aXzaajaaaca GLOaGaayzkaaaacaGLBbGaayzxaaGaeyOKH4QaaGimaaaa@48A7@  en probabilité.

Le théorème 2 est prouvé en annexe.

2.2 Termes de la variance d’ordre plus élevé

Quand V 1 ( θ ^ ) = o ( M 2 / n ) , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOvamaaBa aaleaacaaIXaaabeaakmaabmaabaGafqiUdeNbaKaaaiaawIcacaGL PaaacaaI9aGaam4BamaabmaabaWaaSGbaeaacaWGnbWaaWbaaSqabe aacaaIYaaaaaGcbaGaamOBaaaaaiaawIcacaGLPaaacaGGSaaaaa@404B@ les termes d’ordre plus élevé de la variance sont nécessaires. Le théorème 3 donne ces termes pour le cas particulier de l’échantillonnage aléatoire simple. Sous échantillonnage aléatoire simple, chaque unité est désignée par l’indice inférieur i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAaaaa@355C@ au lieu de h i k . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiAaiaadM gacaWGRbGaaiOlaaaa@37EB@

Théorème 3. Supposons que les conditions (A1) à (A5) sont satisfaites, et qu’un échantillon aléatoire simple de n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaaaa@3561@  unités est tiré de la population de M MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamytaaaa@3540@  unités, où n / M 0. MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSGbaeaaca WGUbaabaGaamytaaaacqGHsgIRcaaIWaGaaiOlaaaa@39A2@  Soit Y ^ c N R = i S w i δ c i y i ( 1 r i ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmywayaaja Waa0baaSqaaiaadogaaeaacaWGobGaamOuaaaakiaai2dadaaeqaqa aiaadEhadaWgaaWcbaGaamyAaaqabaGccqaH0oazdaWgaaWcbaGaam 4yaiaadMgaaeqaaOGaamyEamaaBaaaleaacaWGPbaabeaaaeaacaWG PbGaeyicI4Saam4uaaqab0GaeyyeIuoakmaabmaabaGaaGymaiabgk HiTiaadkhadaWgaaWcbaGaamyAaaqabaaakiaawIcacaGLPaaaaaa@4B4D@  le total estimé pour les non-répondants dans la poststrate c . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4yaiaac6 caaaa@3608@  Supposons que y ¯ c R MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmyEayaara Waa0baaSqaaiaadogaaeaacaWGsbaaaaaa@3770@  est indépendant de M ^ c R MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmytayaaja Waa0baaSqaaiaadogaaeaacaWGsbaaaaaa@373C@  et Y ^ c N R , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmywayaaja Waa0baaSqaaiaadogaaeaacaWGobGaamOuaaaakiaacYcaaaa@38D5@  et que tous les r i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCamaaBa aaleaacaWGPbaabeaaaaa@367F@  sont indépendants les uns des autres et sont indépendants de Z i . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOwamaaBa aaleaacaWGPbaabeaakiaac6caaaa@3723@  Alors,

V 2 ( θ ^ ) = c = 1 C 2 p c 1 p c 2 V [ y ¯ c R Y ¯ c R ] V [ M ^ c R M c R ] + o ( M 2 / n 2 ) . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOvamaaBa aaleaacaaIYaaabeaakmaabmaabaGafqiUdeNbaKaaaiaawIcacaGL PaaacaaI9aWaaabCaeqaleaacaWGJbGaaGypaiaaigdaaeaacaWGdb aaniabggHiLdGcdaWcaaqaaiaaikdacaWGWbWaaSbaaSqaaiaadoga aeqaaOGaeyOeI0IaaGymaaqaaiaadchadaqhaaWcbaGaam4yaaqaai aaikdaaaaaaOGaamOvamaadmaabaGabmyEayaaraWaa0baaSqaaiaa dogaaeaacaWGsbaaaOGaeyOeI0IabmywayaaraWaa0baaSqaaiaado gaaeaacaWGsbaaaaGccaGLBbGaayzxaaGaamOvamaadmaabaGabmyt ayaajaWaa0baaSqaaiaadogaaeaacaWGsbaaaOGaeyOeI0Iaamytam aaDaaaleaacaWGJbaabaGaamOuaaaaaOGaay5waiaaw2faaiabgUca Riaad+gadaqadaqaamaalyaabaGaamytamaaCaaaleqabaGaaGOmaa aaaOqaaiaad6gadaahaaWcbeqaaiaaikdaaaaaaaGccaGLOaGaayzk aaGaaGOlaaaa@620A@

Nous pouvons estimer V 2 ( θ ^ ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOvamaaBa aaleaacaaIYaaabeaakmaabmaabaGafqiUdeNbaKaaaiaawIcacaGL Paaaaaa@398A@ dans un échantillon aléatoire simple par

c = 1 C 2 p ^ c 1 p ^ c 2 s c 2 n c R M c p ^ c ( M M c p ^ c ) n , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaabCaeqale aacaWGJbGaaGypaiaaigdaaeaacaWGdbaaniabggHiLdGcdaWcaaqa aiaaikdaceWGWbGbaKaadaWgaaWcbaGaam4yaaqabaGccqGHsislca aIXaaabaGabmiCayaajaWaa0baaSqaaiaadogaaeaacaaIYaaaaaaa kmaalaaabaGaam4CamaaDaaaleaacaWGJbaabaGaaGOmaaaaaOqaai aad6gadaqhaaWcbaGaam4yaaqaaiaadkfaaaaaaOWaaSaaaeaacaWG nbWaaSbaaSqaaiaadogaaeqaaOGabmiCayaajaWaaSbaaSqaaiaado gaaeqaaOWaaeWaaeaacaWGnbGaeyOeI0IaamytamaaBaaaleaacaWG JbaabeaakiqadchagaqcamaaBaaaleaacaWGJbaabeaaaOGaayjkai aawMcaaaqaaiaad6gaaaGaaGilaaaa@5456@

p ^ c MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmiCayaaja WaaSbaaSqaaiaadogaaeqaaaaa@3687@ est le taux de réponse empirique dans la poststrate c , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4yaiaacY caaaa@3606@ n c R MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBamaaDa aaleaacaWGJbaabaGaamOuaaaaaaa@374D@ est le nombre de répondants dans la poststrate c , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4yaiaacY caaaa@3606@ et s c 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4CamaaDa aaleaacaWGJbaabaGaaGOmaaaaaaa@3737@ est la variance d’échantillon de y MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEaaaa@356C@ pour les répondants dans la poststrate c . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4yaiaac6 caaaa@3608@

En pratique, le terme d’ordre un de la variance estimée en se servant de (2.7) sera généralement non nul, même quand V 1 ( θ ^ ) = 0. MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOvamaaBa aaleaacaaIXaaabeaakmaabmaabaGafqiUdeNbaKaaaiaawIcacaGL PaaacaaI9aGaaGimaiaac6caaaa@3BBC@ Donc, le terme d’ordre un estimé ne peut pas être utilisé pour diagnostiquer si des termes d’ordre plus élevé sont nécessaires. Toutefois, l’expression de la variance en (2.6) implique que le terme V 1 ( θ ^ ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOvamaaBa aaleaacaaIXaaabeaakmaabmaabaGafqiUdeNbaKaaaiaawIcacaGL Paaaaaa@3989@ est suffisamment grand pour que l’approximation d’ordre un soit valide quand toutes les poststrates ont des taux de réponse strictement inférieurs à un et que la variance intra-poststrate est non négligeable.

2.3 Jackknife

L’estimateur de variance par jackknife est défini comme il suit :

V ^ J ( θ ^ ) = g = 1 H n g 1 n g j S g ( θ ^ ( g j ) θ ^ ) 2 , ( 2.8 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOvayaaja WaaSbaaSqaaiaadQeaaeqaaOWaaeWaaeaacuaH4oqCgaqcaaGaayjk aiaawMcaaiaai2dadaaeWbqabSqaaiaadEgacaaI9aGaaGymaaqaai aadIeaa0GaeyyeIuoakmaalaaabaGaamOBamaaBaaaleaacaWGNbaa beaakiabgkHiTiaaigdaaeaacaWGUbWaaSbaaSqaaiaadEgaaeqaaa aakmaaqafabeWcbaGaamOAaiabgIGioprr1ngBPrwtHrhAXaqeguuD JXwAKbstHrhAG8KBLbacfaGae8NeXp1aaSbaaeaacaWGNbaabeaaae qaniabggHiLdGcdaqadaqaaiqbeI7aXzaajaWaaWbaaSqabeaadaqa daqaaiaadEgacaWGQbaacaGLOaGaayzkaaaaaOGaeyOeI0IafqiUde NbaKaaaiaawIcacaGLPaaadaahaaWcbeqaaiaaikdaaaGccaaISaGa aGzbVlaaywW7caaMf8UaaGzbVlaaywW7caGGOaGaaGOmaiaac6caca aI4aGaaiykaaaa@6D85@

θ ^ ( g j ) = Y ^ P S ( g j ) Y ^ S S ( g j ) , Y ^ P S ( g j ) = c = 1 C M c h i k S w h i k ( g j ) r h i k δ c h i k y h i k h i k S w h i k ( g j ) r h i k δ c h i k , Y ^ S S ( g j ) = h i k S w h i k ( g j ) y h i k , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipu0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqbaeaabmGaaa qaaiqbeI7aXzaajaWaaWbaaSqabeaadaqadaqaaiaadEgacaWGQbaa caGLOaGaayzkaaaaaaGcbaGaaGypaiqadMfagaqcamaaDaaaleaaca WGqbGaam4uaaqaamaabmaabaGaam4zaiaadQgaaiaawIcacaGLPaaa aaGccqGHsislceWGzbGbaKaadaqhaaWcbaGaam4uaiaadofaaeaada qadaqaaiaadEgacaWGQbaacaGLOaGaayzkaaaaaOGaaGilaaqaaiqa dMfagaqcamaaDaaaleaacaWGqbGaam4uaaqaamaabmaabaGaam4zai aadQgaaiaawIcacaGLPaaaaaaakeaacaaI9aWaaabCaeaacaWGnbWa aSbaaSqaaiaadogaaeqaaaqaaiaadogacaaI9aGaaGymaaqaaiaado eaa0GaeyyeIuoakmaalaaabaWaaabuaeqaleaacaWGObGaamyAaiaa dUgacqGHiiIZcaWGtbaabeqdcqGHris5aOGaam4DamaaDaaaleaaca WGObGaamyAaiaadUgaaeaadaqadaqaaiaadEgacaWGQbaacaGLOaGa ayzkaaaaaOGaamOCamaaBaaaleaacaWGObGaamyAaiaadUgaaeqaaO GaeqiTdq2aaSbaaSqaaiaadogacaWGObGaamyAaiaadUgaaeqaaOGa amyEamaaBaaaleaacaWGObGaamyAaiaadUgaaeqaaaGcbaWaaabuae qaleaacaWGObGaamyAaiaadUgacqGHiiIZcaWGtbaabeqdcqGHris5 aOGaam4DamaaDaaaleaacaWGObGaamyAaiaadUgaaeaadaqadaqaai aadEgacaWGQbaacaGLOaGaayzkaaaaaOGaamOCamaaBaaaleaacaWG ObGaamyAaiaadUgaaeqaaOGaeqiTdq2aaSbaaSqaaiaadogacaWGOb GaamyAaiaadUgaaeqaaaaakiaaiYcaaeaaceWGzbGbaKaadaqhaaWc baGaam4uaiaadofaaeaadaqadaqaaiaadEgacaWGQbaacaGLOaGaay zkaaaaaaGcbaGaaGypamaaqafabeWcbaGaamiAaiaadMgacaWGRbGa eyicI4Saam4uaaqab0GaeyyeIuoakiaadEhadaqhaaWcbaGaamiAai aadMgacaWGRbaabaWaaeWaaeaacaWGNbGaamOAaaGaayjkaiaawMca aaaakiaadMhadaWgaaWcbaGaamiAaiaadMgacaWGRbaabeaakiaaiY caaaaaaa@A622@

et les poids jackknife sont donnés par :

w h i k ( g j ) = { 0 si ( h i ) = ( g j ) n h n h 1 w h i k si h = g , i j w h i k si h g . ( 2.9 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrVfpC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4DamaaDa aaleaacaWGObGaamyAaiaadUgaaeaadaqadaqaaiaadEgacaWGQbaa caGLOaGaayzkaaaaaOGaaGypamaaceaabaqbaeaabmWaaaqaaiaaic daaeaacaqGZbGaaeyAaaqaamaabmaabaGaamiAaiaadMgaaiaawIca caGLPaaacaaI9aWaaeWaaeaacaWGNbGaamOAaaGaayjkaiaawMcaaa qaamaalaaabaGaamOBamaaBaaaleaacaWGObaabeaaaOqaaiaad6ga daWgaaWcbaGaamiAaaqabaGccqGHsislcaaIXaaaaiaadEhadaWgaa WcbaGaamiAaiaadMgacaWGRbaabeaaaOqaaiaabohacaqGPbaabaGa amiAaiaai2dacaWGNbGaaGilaiaadMgacqGHGjsUcaWGQbaabaGaam 4DamaaBaaaleaacaWGObGaamyAaiaadUgaaeqaaaGcbaGaae4Caiaa bMgaaeaacaWGObGaeyiyIKRaam4zaaaaaiaawUhaaiaai6cacaaMf8 UaaGzbVlaaywW7caaMf8UaaGzbVlaaywW7caGGOaGaaGOmaiaac6ca caaI5aGaaiykaaaa@7172@

Si ( n / M 2 ) V 1 ( θ ^ ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaada Wcgaqaaiaad6gaaeaacaWGnbWaaWbaaSqabeaacaaIYaaaaaaaaOGa ayjkaiaawMcaaiaadAfadaWgaaWcbaGaaGymaaqabaGcdaqadaqaai qbeI7aXzaajaaacaGLOaGaayzkaaaaaa@3DE0@ converge vers une constante positive et que les hypothèses (A1) à (A5) sont vérifiées, alors V ^ J ( θ ^ ) / V 1 ( θ ^ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSGbaeaace WGwbGbaKaadaWgaaWcbaGaamOsaaqabaGcdaqadaqaaiqbeI7aXzaa jaaacaGLOaGaayzkaaaabaGaamOvamaaBaaaleaacaaIXaaabeaakm aabmaabaGafqiUdeNbaKaaaiaawIcacaGLPaaaaaaaaa@3EDD@ converge vers 1 en probabilité. Cela découle des arguments jackknife classiques (théorème 6.1 de Shao et Tu 1995), parce que le paramètre de population est une fonction continûment dérivable des totaux de populations. Sous les conditions du théorème 2, soit θ ^ / V ^ L ( θ ^ ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSGbaeaacu aH4oqCgaqcaaqaamaakaaabaGabmOvayaajaWaaSbaaSqaaiaadYea aeqaaOWaaeWaaeaacuaH4oqCgaqcaaGaayjkaiaawMcaaaWcbeaaaa aaaa@3BA6@ ou θ ^ / V ^ J ( θ ^ ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSGbaeaacu aH4oqCgaqcaaqaamaakaaabaGabmOvayaajaWaaSbaaSqaaiaadQea aeqaaOWaaeWaaeaacuaH4oqCgaqcaaGaayjkaiaawMcaaaWcbeaaaa aaaa@3BA4@ peut être utilisée comme statistique de test. Chacune suit approximativement une loi normale centrée réduite quand l’hypothèse nulle H 0 : θ = 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamisamaaBa aaleaacaaIWaaabeaakiaaykW7caaI6aGaeqiUdeNaaGypaiaaicda aaa@3BB1@ est vérifiée.

2.4 Remarques et extensions

À la présente section, nous avons obtenu l’estimateur de variance par linéarisation pour comparer le total de population estimé d’une quantité connue pour toutes les unités dans l’échantillon sélectionné à l’estimation poststratifiée calculée en utilisant uniquement les répondants. Les théorèmes 1 et 2 donnent aussi la variance et l’estimateur de variance pour comparer l’estimateur calculé en utilisant l’échantillon sélectionné à celui obtenu pour les répondants pondérés par les poids de base. Dans ce cas, Y ^ P S MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmywayaaja WaaSbaaSqaaiaadcfacaWGtbaabeaaaaa@3735@ se réduit à un estimateur avec une poststrate, Y ^ P S = ( M / M ^ R ) Y ^ R , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmywayaaja WaaSbaaSqaaiaadcfacaWGtbaabeaakiaai2dadaqadaqaamaalyaa baGaamytaaqaaiqad2eagaqcamaaCaaaleqabaGaamOuaaaaaaaaki aawIcacaGLPaaaceWGzbGbaKaadaahaaWcbeqaaiaadkfaaaGccaGG Saaaaa@3F13@ M ^ R = ( h i k ) S w h i k r h i k . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmytayaaja WaaWbaaSqabeaacaWGsbaaaOGaaGypamaaqababaGaam4DamaaBaaa leaacaWGObGaamyAaiaadUgaaeqaaOGaamOCamaaBaaaleaacaWGOb GaamyAaiaadUgaaeqaaaqaamaabmaabaGaamiAaiaadMgacaWGRbaa caGLOaGaayzkaaGaeyicI4Saam4uaaqab0GaeyyeIuoakiaac6caaa a@4854@

Que se passe-t-il si y MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEaaaa@356C@ est l’une des variables de poststratification ? Dans le cadre utilisé à la présente section, les chiffres de population pour les variables de poststratification sont tirés de la base de sondage ou d’une source externe. Si y MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEaaaa@356C@ est une combinaison linéaire d’indicateurs de classe de poststratification, alors Y ^ P S MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmywayaaja WaaSbaaSqaaiaadcfacaWGtbaabeaaaaa@3735@ est le même pour tous les échantillons possibles et sa variance est donc nulle. Alors V ( θ ^ ) = V ( Y ^ S S ) , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOvamaabm aabaGafqiUdeNbaKaaaiaawIcacaGLPaaacaaI9aGaamOvamaabmaa baGabmywayaajaWaaSbaaSqaaiaadofacaWGtbaabeaaaOGaayjkai aawMcaaiaacYcaaaa@3F47@ qui est le terme d’ordre un de la variance dans le théorème 1. Si y MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEaaaa@356C@ est aussi une variable de stratification dans le plan de sondage, alors V ( θ ^ ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOvamaabm aabaGafqiUdeNbaKaaaiaawIcacaGLPaaaaaa@3898@ sera nulle. Si y MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEaaaa@356C@ n’est pas une variable de stratification, alors habituellement Y ^ S S MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmywayaaja WaaSbaaSqaaiaadofacaWGtbaabeaaaaa@3738@ variera d’un échantillon à l’autre et aura une variance O ( M 2 / n ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4tamaabm aabaWaaSGbaeaacaWGnbWaaWbaaSqabeaacaaIYaaaaaGcbaGaamOB aaaaaiaawIcacaGLPaaaaaa@3999@ de sorte que le test du biais de non-réponse peut être effectué. Nous nous attendrions à ce que le taux de rejet pour le test corresponde au seuil de signification α MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySdegaaa@360D@ dans ce cas.

Le paramètre θ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiUdehaaa@3624@ dans (2.3) a été défini comme étant la différence entre le total de population poststratifié, calculé en utilisant les propensions à répondre dans la population sous le schéma de poststratification adopté, et le total de population non ajusté. Dans (2.4), le total de population non ajusté Y MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamywaaaa@354C@ a été estimé au moyen de l’estimateur d’Horvitz-Thompson. Le paramètre θ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiUdehaaa@3624@ pourrait aussi être estimé au moyen de l’expression

θ ^ 2 = Y ^ P S c = 1 C M c Y ^ c M ^ c , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiUdeNbaK aadaWgaaWcbaGaaGOmaaqabaGccaaI9aGabmywayaajaWaaSbaaSqa aiaadcfacaWGtbaabeaakiabgkHiTmaaqahabaGaamytamaaBaaale aacaWGJbaabeaaaeaacaWGJbGaaGypaiaaigdaaeaacaWGdbaaniab ggHiLdGcdaWcaaqaaiqadMfagaqcamaaBaaaleaacaWGJbaabeaaaO qaaiqad2eagaqcamaaBaaaleaacaWGJbaabeaaaaGccaaISaaaaa@47D5@

dans laquelle un estimateur poststratifié est utilisé au lieu de Y ^ S S . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmywayaaja WaaSbaaSqaaiaadofacaWGtbaabeaakiaac6caaaa@37F4@ La variance de θ ^ 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiUdeNbaK aadaWgaaWcbaGaaGOmaaqabaaaaa@371C@ devrait, en principe, être inférieure à la variance de θ ^ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiUdeNbaK aaaaa@3634@ sous les hypothèses de poststratification, ce qui donne lieu à un test plus puissant. Cependant, quand y MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEaaaa@356C@ est une combinaison linéaire des indicateurs de poststrate, la statistique θ ^ 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiUdeNbaK aadaWgaaWcbaGaaGOmaaqabaaaaa@371C@ ne peut pas être utilisée pour tester H 0 : θ = 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamisamaaBa aaleaacaaIWaaabeaakiaaykW7caaI6aGaeqiUdeNaaGypaiaaicda aaa@3BB1@ , parce que V ( θ ^ 2 ) = 0. MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOvamaabm aabaGafqiUdeNbaKaadaWgaaWcbaGaaGOmaaqabaaakiaawIcacaGL PaaacaaI9aGaaGimaiaac6caaaa@3BBD@ Un problème similaire peut se poser quand la variable y MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEaaaa@356C@ est fortement corrélée aux variables de poststratification. Par contre, l’estimateur θ ^ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiUdeNbaK aaaaa@3634@ possède habituellement une variance positive même si y MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEaaaa@356C@ est l’une des variables de poststratification.

Parfois, la poststratification est effectuée en utilisant des totaux de poststratification moins que parfaits  MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGWj0Jf9crFfpeea0xh9v8qiW7rqqrpu0xh9Wqpm0db9Wq pepeuf0xe9q8qiYRWFGCk9vi=dbvc9s8vr0db9Fn0dbbG8Fq0Jfr=x fr=xfbpdbiqaaeaaciGaaiaabeqaamaabaabaaGcbaacbaqcLbwaqa aaaaaaaaWdbiaa=nbiaaa@3D01@ par exemple, les totaux peuvent provenir d’une grande enquête telle que l’American Community Survey qui comporte ses propres erreurs d’échantillonnage et non dues à l’échantillonnage, ou ils peuvent provenir d’un recensement portant sur une population légèrement différente. Dans certains cas, il arrive que des variables de poststratification, telles que la race ou le groupe ethnique, soient mesurées différemment dans l’enquête et dans la source externe des totaux de population. L’utilisation de θ ^ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiUdeNbaK aaaaa@3634@ plutôt que θ ^ 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiUdeNbaK aadaWgaaWcbaGaaGOmaaqabaaaaa@371C@ pourrait déceler des différences éventuellement  causées par une mauvaise poststratification.

Au besoin, les tests peuvent être effectués en utilisant des moyennes plutôt que des totaux. Dans ce cas, le paramètre de population est

θ M = c = 1 C M c M Y c R M c R Y ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiUde3aaS baaSqaaiaad2eaaeqaaOGaaGypamaaqahabeWcbaGaam4yaiaai2da caaIXaaabaGaam4qaaqdcqGHris5aOWaaSaaaeaacaWGnbWaaSbaaS qaaiaadogaaeqaaaGcbaGaamytaaaadaWcaaqaaiaadMfadaqhaaWc baGaam4yaaqaaiaadkfaaaaakeaacaWGnbWaa0baaSqaaiaadogaae aacaWGsbaaaaaakiabgkHiTiqadMfagaqeaaaa@47D2@

Y ¯ = Y / M , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmywayaara GaaGypamaalyaabaGaamywaaqaaiaad2eaaaGaaiilaaaa@38A1@ et peut être estimé par

θ ^ M = c = 1 C M c M Y ^ c R M ^ c R Y ^ S S h i k S w h i k . ( 2.10 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiUdeNbaK aadaWgaaWcbaGaamytaaqabaGccaaI9aWaaabCaeqaleaacaWGJbGa aGypaiaaigdaaeaacaWGdbaaniabggHiLdGcdaWcaaqaaiaad2eada WgaaWcbaGaam4yaaqabaaakeaacaWGnbaaamaalaaabaGabmywayaa jaWaa0baaSqaaiaadogaaeaacaWGsbaaaaGcbaGabmytayaajaWaa0 baaSqaaiaadogaaeaacaWGsbaaaaaakiabgkHiTmaalaaabaGabmyw ayaajaWaaSbaaSqaaiaadofacaWGtbaabeaaaOqaamaaqafabeWcba GaamiAaiaadMgacaWGRbGaeyicI4Saam4uaaqab0GaeyyeIuoakiaa dEhadaWgaaWcbaGaamiAaiaadMgacaWGRbaabeaaaaGccaaIUaGaaG zbVlaaywW7caaMf8UaaGzbVlaaywW7caGGOaGaaGOmaiaac6cacaaI XaGaaGimaiaacMcaaaa@61FC@

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