Tests pour évaluer le biais de non-réponse dans les enquêtes
Section 2. PoststratificationTests 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@
où
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@
où
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.
(A1) Le nombre de poststrates,
C
,
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9
qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr
0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4qaiaacY
caaaa@35E6@
est fixé et
M
c
/
M
→
λ
c
∈
(
0,1
)
.
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9
qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr
0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSGbaeaaca
WGnbWaaSbaaSqaaiaadogaaeqaaaGcbaGaamytaaaacqGHsgIRcqaH
7oaBdaWgaaWcbaGaam4yaaqabaGccqGHiiIZdaqadaqaaiaaicdaca
aISaGaaGymaaGaayjkaiaawMcaaiaac6caaaa@41EF@
(A2) Il existe une constante
K
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9
qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr
0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4saaaa@353E@
telle que
|
y
h
i
k
|
<
K
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9
qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr
0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaqWaaeaaca
aMc8UaamyEamaaBaaaleaacaWGObGaamyAaiaadUgaaeqaaOGaaGPa
VdGaay5bSlaawIa7aiaaiYdacaWGlbaaaa@403B@
pour toute unité
(
h
i
k
)
.
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9
qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr
0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaaca
WGObGaamyAaiaadUgaaiaawIcacaGLPaaacaGGUaaaaa@3974@
(A3)
max
h
i
k
w
h
i
k
=
O
(
M
/
n
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9
qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr
0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciyBaiaacg
gacaGG4bWaaSbaaSqaaiaadIgacaWGPbGaam4AaaqabaGccaWG3bWa
aSbaaSqaaiaadIgacaWGPbGaam4AaaqabaGccaaI9aGaam4tamaabm
aabaWaaSGbaeaacaWGnbaabaGaamOBaaaaaiaawIcacaGLPaaaaaa@433F@
et
max
h
i
k
w
h
i
k
/
w
h
i
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9
qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr
0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciyBaiaacg
gacaGG4bWaaSbaaSqaaiaadIgacaWGPbGaam4AaaqabaGcdaWcgaqa
aiaadEhadaWgaaWcbaGaamiAaiaadMgacaWGRbaabeaaaOqaaiaadE
hadaWgaaWcbaGaamiAaiaadMgaaeqaaaaaaaa@4159@
est borné.
(A4)
R
h
i
k
>
ε
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9
qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr
0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOuamaaBa
aaleaacaWGObGaamyAaiaadUgaaeqaaOGaaGOpaiabew7aLbaa@3AB5@
pour toute unité
(
h
i
k
)
,
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9
qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr
0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaaca
WGObGaamyAaiaadUgaaiaawIcacaGLPaaacaGGSaaaaa@3972@
pour
ε
>
0
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9
qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr
0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyTduMaaG
Opaiaaicdaaaa@3797@
fixe. Cela garantit que
chaque unité possède une propension à répondre positive qui possède une borne
strictement non nulle.
(A5) Le vecteur des indicateurs
de réponse
r
=
[
r
h
i
k
]
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9
qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr
0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCOCaiaai2
dadaWadaqaaiaadkhadaWgaaWcbaGaamiAaiaadMgacaWGRbaabeaa
aOGaay5waiaaw2faaaaa@3C1A@
est indépendant du vecteur des
indicateurs d’inclusion dans l’échantillon
Z
=
[
Z
h
i
k
]
.
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9
qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr
0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCOwaiaai2
dadaWadaqaaiaadQfadaWgaaWcbaGaamiAaiaadMgacaWGRbaabeaa
aOGaay5waiaaw2faaiaac6caaaa@3C9C@
En outre,
r
h
i
k
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9
qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr
0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCamaaBa
aaleaacaWGObGaamyAaiaadUgaaeqaaaaa@385C@
et
r
l
j
p
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9
qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr
0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCamaaBa
aaleaacaWGSbGaamOAaiaadchaaeqaaaaa@3866@
sont indépendants quand
(
h
i
)
≠
(
l
j
)
,
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqr=fFD0xd9Wqpe0dd9
qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr
0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaaca
WGObGaamyAaaGaayjkaiaawMcaaiabgcMi5oaabmaabaGaamiBaiaa
dQgaaiaawIcacaGLPaaacaGGSaaaaa@3DB2@
de sorte que les indicateurs
de réponse dans les diverses UPE ne sont pas corrélés.
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@
où
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@
où
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@
où
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@
où
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@
où
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@
où
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@
ISSN : 1712-5685
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Techniques d ’enquête publie des articles sur les divers aspects des méthodes statistiques qui intéressent un organisme statistique comme, par exemple, les problèmes de conception découlant de contraintes d’ordre pratique, l’utilisation de différentes sources de données et de méthodes de collecte, les erreurs dans les enquêtes, l’évaluation des enquêtes, la recherche sur les méthodes d’enquête, l’analyse des séries chronologiques, la désaisonnalisation, les études démographiques, l’intégration de données statistiques, les méthodes d’estimation et d’analyse de données et le développement de systèmes généralisés. Une importance particulière est accordée à l’élaboration et à l’évaluation de méthodes qui ont été utilisées pour la collecte de données ou appliquées à des données réelles. Tous les articles seront soumis à une critique, mais les auteurs demeurent responsables du contenu de leur texte et les opinions émises dans la revue ne sont pas nécessairement celles du comité de rédaction ni de Statistique Canada.
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Droit d'auteur
Publication autorisée par le ministre responsable de Statistique Canada.
© Ministre de l'Industrie, 2016
L'utilisation de la présente publication est assujettie aux modalités de l'Entente de licence ouverte de Statistique Canada .
N° 12-001-X au catalogue
Périodicité : Semi-annuel
Ottawa
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Date de modification :
2016-12-20