Estimation de la variance dans le calage à plusieurs phases
Section 5. Conclusion

Le présent article décrit la construction d’une nouvelle représentation des poids calés en plusieurs phases qui permet de représenter un estimateur calé en plusieurs phases sous la forme d’un estimateur multivarié calé en une phase. Cette représentation rend possible le calcul d’une approximation sous forme explicite de la variance des estimateurs calés en plusieurs phases pour tout nombre de phases. Une comparaison avec une autre approximation connue dans la littérature pour le cas à deux phases montre que, même si les deux approximations sont convergentes, elles diffèrent en ce qui concerne leurs estimations, leur forme et leur interprétation. Nous avons discuté de certains avantages de la nouvelle approximation dans le cas du calage à deux phases et avons aussi montré sa convergence au moyen d’une étude en simulation du calage à trois phases où elle a donné de très bons résultats pour tous les plans étudiés. L’examen de l’efficacité de l’estimateur proposé en fonction des taux d’échantillonnage et d’autres paramètres du plan fera l’objet de futurs travaux de recherche.

Annexe A

Pour abréger la notation, nous effectuerons notre analyse sous forme matricielle. Nous utiliserons la convention selon laquelle, pour j > i , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOAaiaai6 dacaWGPbGaaiilaaaa@374F@ la sommation dans les produits scalaires X i w j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiwamaaDa aaleaacaWGPbaabaGccWaGyBOmGikaaiaadEhadaWgaaWcbaGaamOA aaqabaaaaa@3AF3@ et X i D j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiwamaaDa aaleaacaWGPbaabaGccWaGyBOmGikaaiaadseadaWgaaWcbaGaamOA aaqabaaaaa@3AC0@ (ou avec w j * MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4DamaaDa aaleaacaWGQbaabaGaaiOkaaaaaaa@36C0@ ou w ˜ j ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabm4Dayaaia WaaSbaaSqaaiaadQgaaeqaaOGaaiykaaaa@36D7@ est faite sur les unités k s j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4AaiabgI GiolaadohadaWgaaWcbaGaamOAaaqabaaaaa@3881@ (et non sur s i ) , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4CamaaBa aaleaacaWGPbaabeaakiaacMcacaGGSaaaaa@3773@ c’est-à-dire sur l’échantillon indiqué par le dernier ensemble de poids dans le produit scalaire. D’où, Z ^ i j = ( X i D i * X i ) 1 X i ( D j * D j 1 * ) X j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOwayaaja WaaSbaaSqaaiaadMgacaWGQbaabeaakiaai2dadaqadeqaaiaadIfa daqhaaWcbaGaamyAaaqaaOGamai2gkdiIcaacaWGebWaa0baaSqaai aadMgaaeaacaGGQaaaaOGaamiwamaaBaaaleaacaWGPbaabeaaaOGa ayjkaiaawMcaamaaCaaaleqabaGaeyOeI0IaaGymaaaakiaadIfada qhaaWcbaGaamyAaaqaaOGamai2gkdiIcaadaqadeqaaiaadseadaqh aaWcbaGaamOAaaqaaiaacQcaaaGccqGHsislcaWGebWaa0baaSqaai aadQgacqGHsislcaaIXaaabaGaaiOkaaaaaOGaayjkaiaawMcaaiaa dIfadaWgaaWcbaGaamOAaaqabaaaaa@54DE@ sous cette notation.

Preuve du lemme 3.1. Les poids qui satisfont l’équation de calage à la j e MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOAamaaCa aaleqabaGaaeyzaaaaaaa@361E@ phase avec les poids initiaux w ˜ j 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabm4Dayaaia WaaSbaaSqaaiaadQgacqGHsislcaaIXaaabeaaaaa@37C8@ sont donnés par l’équation (3.4). Sous notre notation matricielle

w ˜ j = D j * [ g 1 + + g j ( j 1 ) ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabm4Dayaaia WaaSbaaSqaaiaadQgaaeqaaOGaaGypaiaadseadaqhaaWcbaGaamOA aaqaaiaacQcaaaGcdaWadaqaaiaadEgadaWgaaWcbaGaaGymaaqaba GccqGHRaWkcqWIMaYscqGHRaWkcaWGNbWaaSbaaSqaaiaadQgaaeqa aOGaeyOeI0YaaeWaaeaacaWGQbGaeyOeI0IaaGymaaGaayjkaiaawM caaaGaay5waiaaw2faaaaa@4760@

g j = 1 + X j T j 1 ( X j w ˜ j 1 X j D j w ˜ j 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4zamaaBa aaleaacaWGQbaabeaakiaai2dacaaIXaGaey4kaSIaamiwamaaBaaa leaacaWGQbaabeaakiaadsfadaqhaaWcbaGaamOAaaqaaiabgkHiTi aaigdaaaGcdaqadaqaaiaadIfadaqhaaWcbaGaamOAaaqaaOGamai2 gkdiIcaaceWG3bGbaGaadaWgaaWcbaGaamOAaiabgkHiTiaaigdaae qaaOGaeyOeI0IaamiwamaaDaaaleaacaWGQbaabaGccWaGyBOmGika aiaadseadaWgaaWcbaGaamOAaaqabaGcceWG3bGbaGaadaWgaaWcba GaamOAaiabgkHiTiaaigdaaeqaaaGccaGLOaGaayzkaaaaaa@53F2@ (voir l’équation (3.5)). Donc

w ˜ j = D j 1 * D j [ g 1 + + g j 1 ( j 2 ) + g j 1 ] = D j [ w ˜ j 1 + D j 1 * ( g j 1 ) ] . ( A .1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFjpu0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqbaeaabiGaaa qaaiqadEhagaacamaaBaaaleaacaWGQbaabeaaaOqaaiaai2dacaWG ebWaa0baaSqaaiaadQgacqGHsislcaaIXaaabaGaaiOkaaaakiaads eadaWgaaWcbaGaamOAaaqabaGcdaWadaqaaiaadEgadaWgaaWcbaGa aGymaaqabaGccqGHRaWkcqWIMaYscqGHRaWkcaWGNbWaaSbaaSqaai aadQgacqGHsislcaaIXaaabeaakiabgkHiTmaabmaabaGaamOAaiab gkHiTiaaikdaaiaawIcacaGLPaaacqGHRaWkcaWGNbWaaSbaaSqaai aadQgaaeqaaOGaeyOeI0IaaGymaaGaay5waiaaw2faaaqaaaqaaiaa i2dacaWGebWaaSbaaSqaaiaadQgaaeqaaOWaamWaaeaaceWG3bGbaG aadaWgaaWcbaGaamOAaiabgkHiTiaaigdaaeqaaOGaey4kaSIaamir amaaDaaaleaacaWGQbGaeyOeI0IaaGymaaqaaiaacQcaaaGcdaqada qaaiaadEgadaWgaaWcbaGaamOAaaqabaGccqGHsislcaaIXaaacaGL OaGaayzkaaaacaGLBbGaayzxaaGaaGOlaiaaywW7caaMf8UaaGzbVl aaywW7caaMf8UaaGzbVlaaywW7caaMf8UaaGzbVlaacIcacaqGbbGa aeOlaiaabgdacaGGPaaaaaaa@7629@

L’insertion de g j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4zamaaBa aaleaacaWGQbaabeaaaaa@3601@ donne w ˜ j = D j [ w ˜ j 1 + D j 1 * X j T j 1 ( X j w ˜ j 1 X j D j w ˜ j 1 ) ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabm4Dayaaia WaaSbaaSqaaiaadQgaaeqaaOGaaGypaiaadseadaWgaaWcbaGaamOA aaqabaGcdaWadeqaaiqadEhagaacamaaBaaaleaacaWGQbGaeyOeI0 IaaGymaaqabaGccqGHRaWkcaWGebWaa0baaSqaaiaadQgacqGHsisl caaIXaaabaGaaiOkaaaakiaadIfadaWgaaWcbaGaamOAaaqabaGcca WGubWaa0baaSqaaiaadQgaaeaacqGHsislcaaIXaaaaOWaaeWabeaa caWGybWaa0baaSqaaiaadQgaaeaakiadaITHYaIOaaGabm4Dayaaia WaaSbaaSqaaiaadQgacqGHsislcaaIXaaabeaakiabgkHiTiaadIfa daqhaaWcbaGaamOAaaqaaOGamai2gkdiIcaacaWGebWaaSbaaSqaai aadQgaaeqaaOGabm4DayaaiaWaaSbaaSqaaiaadQgacqGHsislcaaI XaaabeaaaOGaayjkaiaawMcaaaGaay5waiaaw2faaaaa@5F55@ qui fait intervenir le poids w ˜ j 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabm4Dayaaia WaaSbaaSqaaiaadQgacqGHsislcaaIXaaabeaaaaa@37C8@ provenant de la phase de calage précédente et son produit scalaire avec X j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiwamaaDa aaleaacaWGQbaabaGccWaGyBOmGikaaaaa@38DD@ et X j D j , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiwamaaDa aaleaacaWGQbaabaGccWaGyBOmGikaaiaadseadaWgaaWcbaGaamOA aaqabaGccaaMb8Uaaiilaaaa@3D05@ tandis que les autres multiplicateurs sont des paramètres du plan. L’expression entre crochets contient trois opérandes et donc, après j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOAaaaa@34E9@ phases de calage, nous aurions 3 j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaG4mamaaCa aaleqabaGaamOAaaaaaaa@35D3@ opérandes qui contiendraient uniquement des paramètres du plan. L’introduction de w ˜ j 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabm4Dayaaia WaaSbaaSqaaiaadQgacqGHsislcaaIXaaabeaaaaa@37C8@ provenant de (A.1) par substitution dans X j D j w ˜ j 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiwamaaDa aaleaacaWGQbaabaGccWaGyBOmGikaaiaadseadaWgaaWcbaGaamOA aaqabaGcceWG3bGbaGaadaWgaaWcbaGaamOAaiabgkHiTiaaigdaae qaaaaa@3E99@ donne

X j D j w ˜ j 1 = X j D j { D j 1 w ˜ j 2 + D j 1 * ( g j 1 1 ) } = X j D j D j 1 w ˜ j 2 + X j D j D j 1 * X j 1 T j 1 1 ( X j 1 w ˜ j 2 X j 1 D j 1 w ˜ j 2 ) ( A .2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFjpu0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqbaeaabiGaaa qaaiaadIfadaqhaaWcbaGaamOAaaqaaOGamai2gkdiIcaacaWGebWa aSbaaSqaaiaadQgaaeqaaOGabm4DayaaiaWaaSbaaSqaaiaadQgacq GHsislcaaIXaaabeaaaOqaaiaai2dacaWGybWaa0baaSqaaiaadQga aeaakiadaITHYaIOaaGaamiramaaBaaaleaacaWGQbaabeaakmaacm aabaGaamiramaaBaaaleaacaWGQbGaeyOeI0IaaGymaaqabaGcceWG 3bGbaGaadaWgaaWcbaGaamOAaiabgkHiTiaaikdaaeqaaOGaey4kaS IaamiramaaDaaaleaacaWGQbGaeyOeI0IaaGymaaqaaiaacQcaaaGc daqadaqaaiaadEgadaWgaaWcbaGaamOAaiabgkHiTiaaigdaaeqaaO GaeyOeI0IaaGymaaGaayjkaiaawMcaaaGaay5Eaiaaw2haaaqaaaqa aiaai2dacaWGybWaa0baaSqaaiaadQgaaeaakiadaITHYaIOaaGaam iramaaBaaaleaacaWGQbaabeaakiaadseadaWgaaWcbaGaamOAaiab gkHiTiaaigdaaeqaaOGabm4DayaaiaWaaSbaaSqaaiaadQgacqGHsi slcaaIYaaabeaakiabgUcaRiaadIfadaqhaaWcbaGaamOAaaqaaOGa mai2gkdiIcaacaWGebWaaSbaaSqaaiaadQgaaeqaaOGaamiramaaDa aaleaacaWGQbGaeyOeI0IaaGymaaqaaiaacQcaaaGccaWGybWaaSba aSqaaiaadQgacqGHsislcaaIXaaabeaakiaadsfadaqhaaWcbaGaam OAaiabgkHiTiaaigdaaeaacqGHsislcaaIXaaaaOWaaeWaaeaacaWG ybWaa0baaSqaaiaadQgacqGHsislcaaIXaaabaGccWaGyBOmGikaai qadEhagaacamaaBaaaleaacaWGQbGaeyOeI0IaaGOmaaqabaGccqGH sislcaWGybWaa0baaSqaaiaadQgacqGHsislcaaIXaaabaGccWaGyB OmGikaaiaadseadaWgaaWcbaGaamOAaiabgkHiTiaaigdaaeqaaOGa bm4DayaaiaWaaSbaaSqaaiaadQgacqGHsislcaaIYaaabeaaaOGaay jkaiaawMcaaiaaywW7caaMf8UaaGzbVlaaywW7caaMf8Uaaiikaiaa bgeacaqGUaGaaeOmaiaacMcaaaaaaa@A5CF@

et donc aussi

X j w ˜ j 1 = X j D j 1 w ˜ j 2 + X j D j 1 * X j 1 T j 1 1 ( X j 1 w ˜ j 2 X j 1 D j 1 w ˜ j 2 ) . ( A .3 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiwamaaDa aaleaacaWGQbaabaGcdaahaaadbeqaaKqzGfGamai2gkdiIcaaaaGc ceWG3bGbaGaadaWgaaWcbaGaamOAaiabgkHiTiaaigdaaeqaaOGaaG ypaiaadIfadaqhaaWcbaGaamOAaaqaaOGamai2gkdiIcaacaWGebWa aSbaaSqaaiaadQgacqGHsislcaaIXaaabeaakiqadEhagaacamaaBa aaleaacaWGQbGaeyOeI0IaaGOmaaqabaGccqGHRaWkcaWGybWaa0ba aSqaaiaadQgaaeaakiadaITHYaIOaaGaamiramaaDaaaleaacaWGQb GaeyOeI0IaaGymaaqaaiaacQcaaaGccaWGybWaaSbaaSqaaiaadQga cqGHsislcaaIXaaabeaakiaadsfadaqhaaWcbaGaamOAaiabgkHiTi aaigdaaeaacqGHsislcaaIXaaaaOWaaeWaaeaacaWGybWaa0baaSqa aiaadQgacqGHsislcaaIXaaabaGccWaGyBOmGikaaiqadEhagaacam aaBaaaleaacaWGQbGaeyOeI0IaaGOmaaqabaGccqGHsislcaWGybWa a0baaSqaaiaadQgacqGHsislcaaIXaaabaGccWaGyBOmGikaaiaads eadaWgaaWcbaGaamOAaiabgkHiTiaaigdaaeqaaOGabm4DayaaiaWa aSbaaSqaaiaadQgacqGHsislcaaIYaaabeaaaOGaayjkaiaawMcaai aai6cacaaMf8UaaGzbVlaaywW7caaMf8UaaiikaiaabgeacaqGUaGa ae4maiaacMcaaaa@831F@

La combinaison des termes donne une expression pour w ˜ j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabm4Dayaaia WaaSbaaSqaaiaadQgaaeqaaaaa@3620@ qui fait intervenir les poids calés provenant de la phase j 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOAaiabgk HiTiaaikdaaaa@3692@ uniquement

w ˜ j = D j D j 1 w ˜ j 2 + D j * X j 1 T j 1 1 ( X j 1 w ˜ j 2 X j 1 D j 1 w ˜ j 2 ) + D j * X j T j 1 ( X j D j 1 w ˜ j 2 X j D j D j 1 w ˜ j 2 ) D j * X j T j 1 Z ^ j 1 , j ( X j 1 w ˜ j 2 X j 1 D j 1 w ˜ j 2 ) . ( A .4 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFjpu0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqbaeaabqGaaa aabaGabm4DayaaiaWaaSbaaSqaaiaadQgaaeqaaaGcbaGaaGypaiaa dseadaWgaaWcbaGaamOAaaqabaGccaWGebWaaSbaaSqaaiaadQgacq GHsislcaaIXaaabeaakiqadEhagaacamaaBaaaleaacaWGQbGaeyOe I0IaaGOmaaqabaaakeaaaeaacaaMc8UaaGPaVlabgUcaRiaadseada qhaaWcbaGaamOAaaqaaiaacQcaaaGccaWGybWaaSbaaSqaaiaadQga cqGHsislcaaIXaaabeaakiaadsfadaqhaaWcbaGaamOAaiabgkHiTi aaigdaaeaacqGHsislcaaIXaaaaOWaaeWaaeaacaWGybWaa0baaSqa aiaadQgacqGHsislcaaIXaaabaGccWaGyBOmGikaaiqadEhagaacam aaBaaaleaacaWGQbGaeyOeI0IaaGOmaaqabaGccqGHsislcaWGybWa a0baaSqaaiaadQgacqGHsislcaaIXaaabaGccWaGyBOmGikaaiaads eadaWgaaWcbaGaamOAaiabgkHiTiaaigdaaeqaaOGabm4DayaaiaWa aSbaaSqaaiaadQgacqGHsislcaaIYaaabeaaaOGaayjkaiaawMcaaa qaaaqaaiaaykW7caaMc8Uaey4kaSIaamiramaaDaaaleaacaWGQbaa baGaaiOkaaaakiaadIfadaWgaaWcbaGaamOAaaqabaGccaWGubWaa0 baaSqaaiaadQgaaeaacqGHsislcaaIXaaaaOWaaeWaaeaacaWGybWa a0baaSqaaiaadQgaaeaakiadaITHYaIOaaGaamiramaaBaaaleaaca WGQbGaeyOeI0IaaGymaaqabaGcceWG3bGbaGaadaWgaaWcbaGaamOA aiabgkHiTiaaikdaaeqaaOGaeyOeI0IaamiwamaaDaaaleaacaWGQb aabaGccWaGyBOmGikaaiaadseadaWgaaWcbaGaamOAaaqabaGccaWG ebWaaSbaaSqaaiaadQgacqGHsislcaaIXaaabeaakiqadEhagaacam aaBaaaleaacaWGQbGaeyOeI0IaaGOmaaqabaaakiaawIcacaGLPaaa aeaaaeaacaaMc8UaaGPaVlabgkHiTiaadseadaqhaaWcbaGaamOAaa qaaiaacQcaaaGccaWGybWaaSbaaSqaaiaadQgaaeqaaOGaamivamaa DaaaleaacaWGQbaabaGaeyOeI0IaaGymaaaakiqadQfagaqcamaaDa aaleaacaWGQbGaeyOeI0IaaGymaiaacYcacaaMc8UaamOAaaqaaOGa mai2gkdiIcaadaqadaqaaiaadIfadaqhaaWcbaGaamOAaiabgkHiTi aaigdaaeaakiadaITHYaIOaaGabm4DayaaiaWaaSbaaSqaaiaadQga cqGHsislcaaIYaaabeaakiabgkHiTiaadIfadaqhaaWcbaGaamOAai abgkHiTiaaigdaaeaakiadaITHYaIOaaGaamiramaaBaaaleaacaWG QbGaeyOeI0IaaGymaaqabaGcceWG3bGbaGaadaWgaaWcbaGaamOAai abgkHiTiaaikdaaeqaaaGccaGLOaGaayzkaaGaaGOlaiaaywW7caaM f8UaaGzbVlaaywW7caaMf8UaaiikaiaabgeacaqGUaGaaeinaiaacM caaaaaaa@D0B1@

L’insertion de (A.2) et (A.3) avec j = p MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOAaiaai2 dacaWGWbaaaa@36A5@ dans (A.1) et la récurrence p 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCaiabgk HiTiaaigdaaaa@3697@ fois sur les groupes de calage respectifs produisent le résultat souhaité.

Annexe B

Un estimateur convergent du total de population dans le calage à trois phases peut être représenté par w ^ 3  ′ y = Y ^ HT 3 + i = 1 3 ( t ^ 1 t ^ 1 + ) γ ^ i , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabm4Dayaaja Waa0baaSqaaiaaiodaaeaakiadaITHYaIOaaGaamyEaiaai2daceWG zbGbaKaadaWgaaWcbaGaaeisaiaabsfadaWgaaadbaGaaG4maaqaba aaleqaaOGaey4kaSYaaabmaeqaleaacaWGPbGaaGypaiaaigdaaeaa caaIZaaaniabggHiLdGcdaqadaqaaiqadshagaqcamaaDaaaleaaca aIXaaabaGaeyOeI0caaOGaeyOeI0IabmiDayaajaWaa0baaSqaaiaa igdaaeaacqGHRaWkaaaakiaawIcacaGLPaaadaahaaWcbeqaaOGama i2gkdiIcaacuaHZoWzgaqcamaaBaaaleaacaWGPbaabeaakiaaygW7 caGGSaaaaa@54DC@

γ ^ 1 = β ^ 1 Z ^ 12 β ^ 2 Z ^ 13 β ^ 3 + Z ^ 12 Z ^ 23 β ^ 3 γ ^ 2 = β ^ 2 Z ^ 23 β ^ 3 γ ^ 3 = β ^ 3 . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqbaeaabmGaaa qaaiqbeo7aNzaajaWaaSbaaSqaaiaaigdaaeqaaaGcbaGaaGypaiqb ek7aIzaajaWaaSbaaSqaaiaaigdaaeqaaOGaeyOeI0IabmOwayaaja WaaSbaaSqaaiaaigdacaaIYaaabeaakiqbek7aIzaajaWaaSbaaSqa aiaaikdaaeqaaOGaeyOeI0IabmOwayaajaWaaSbaaSqaaiaaigdaca aIZaaabeaakiqbek7aIzaajaWaaSbaaSqaaiaaiodaaeqaaOGaey4k aSIabmOwayaajaWaaSbaaSqaaiaaigdacaaIYaaabeaakiqadQfaga qcamaaBaaaleaacaaIYaGaaG4maaqabaGccuaHYoGygaqcamaaBaaa leaacaaIZaaabeaaaOqaaiqbeo7aNzaajaWaaSbaaSqaaiaaikdaae qaaaGcbaGaaGypaiqbek7aIzaajaWaaSbaaSqaaiaaikdaaeqaaOGa eyOeI0IabmOwayaajaWaaSbaaSqaaiaaikdacaaIZaaabeaakiqbek 7aIzaajaWaaSbaaSqaaiaaiodaaeqaaaGcbaGafq4SdCMbaKaadaWg aaWcbaGaaG4maaqabaaakeaacaaI9aGafqOSdiMbaKaadaWgaaWcba GaaG4maaqabaGccaaIUaaaaaaa@6245@

Un estimateur convergent de la variance est donné par

V ^ P ( w ˜ 3  ′ y ) = k , l s 1 ( w 1 k * w 1 l * w 1 k l * ) e ^ 1 k e ^ 1 l + + k , l s 3 ( w 3 k * w 3 l * w 3 k l * ) e ^ 3 k e ^ 3 l + 2 k s 1 , l s 2 w 2 l ( w 1 k w 1 l w 1 k l ) e ^ 1 k e ^ 2 l + 2 k s 2 , l s 3 w 3 l ( w 2 k * w 2 l * w 2 k l * ) e ^ 2 k e ^ 3 l + 2 k s 1 , l s 3 w 2 l w 3 l ( w 3 k * w 3 l * w 3 k l * ) e ^ 1 k e ^ 3 l . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqbaeaabmGaaa qaaiqadAfagaqcamaaBaaaleaacaWGqbaabeaakmaabmaabaGabm4D ayaaiaWaa0baaSqaaiaaiodaaeaakiadaITHYaIOaaGaamyEaaGaay jkaiaawMcaaaqaaiaai2dadaaeqbqabSqaaiaadUgacaaMb8UaaGil aiaaysW7caWGSbGaeyicI4Saam4CamaaBaaameaacaaIXaaabeaaaS qab0GaeyyeIuoakmaabmaabaGaam4DamaaDaaaleaacaaIXaGaam4A aaqaaiaacQcaaaGccaWG3bWaa0baaSqaaiaaigdacaWGSbaabaGaai OkaaaakiabgkHiTiaadEhadaqhaaWcbaGaaGymaiaadUgacaWGSbaa baGaaiOkaaaaaOGaayjkaiaawMcaaiqadwgagaqcamaaBaaaleaaca aIXaGaam4AaaqabaGcceWGLbGbaKaadaWgaaWcbaGaaGymaiaadYga aeqaaOGaey4kaSIaeSOjGSKaey4kaSYaaabuaeqaleaacaWGRbGaaG zaVlaaiYcacaaMc8UaamiBaiabgIGiolaadohadaWgaaadbaGaaG4m aaqabaaaleqaniabggHiLdGcdaqadaqaaiaadEhadaqhaaWcbaGaaG 4maiaadUgaaeaacaGGQaaaaOGaam4DamaaDaaaleaacaaIZaGaamiB aaqaaiaacQcaaaGccqGHsislcaWG3bWaa0baaSqaaiaaiodacaWGRb GaamiBaaqaaiaacQcaaaaakiaawIcacaGLPaaaceWGLbGbaKaadaWg aaWcbaGaaG4maiaadUgaaeqaaOGabmyzayaajaWaaSbaaSqaaiaaio dacaWGSbaabeaaaOqaaaqaaiaaykW7caaMc8Uaey4kaSIaaGOmamaa qafabeWcbaGaam4AaiabgIGiolaadohadaWgaaadbaGaaGymaaqaba WccaaMb8UaaGilaiaaysW7caWGSbGaeyicI4Saam4CamaaBaaameaa caaIYaaabeaaaSqab0GaeyyeIuoakiaadEhadaWgaaWcbaGaaGOmai aadYgaaeqaaOWaaeWaaeaacaWG3bWaaSbaaSqaaiaaigdacaWGRbaa beaakiaadEhadaWgaaWcbaGaaGymaiaadYgaaeqaaOGaeyOeI0Iaam 4DamaaBaaaleaacaaIXaGaam4AaiaadYgaaeqaaaGccaGLOaGaayzk aaGabmyzayaajaWaaSbaaSqaaiaaigdacaWGRbaabeaakiqadwgaga qcamaaBaaaleaacaaIYaGaamiBaaqabaGccqGHRaWkcaaIYaWaaabu aeqaleaacaWGRbGaeyicI4Saam4CamaaBaaameaacaaIYaaabeaali aaygW7caaISaGaaGjbVlaadYgacqGHiiIZcaWGZbWaaSbaaWqaaiaa iodaaeqaaaWcbeqdcqGHris5aOGaam4DamaaBaaaleaacaaIZaGaam iBaaqabaGcdaqadaqaaiaadEhadaqhaaWcbaGaaGOmaiaadUgaaeaa caGGQaaaaOGaam4DamaaDaaaleaacaaIYaGaamiBaaqaaiaacQcaaa GccqGHsislcaWG3bWaa0baaSqaaiaaikdacaWGRbGaamiBaaqaaiaa cQcaaaaakiaawIcacaGLPaaaceWGLbGbaKaadaWgaaWcbaGaaGOmai aadUgaaeqaaOGabmyzayaajaWaaSbaaSqaaiaaiodacaWGSbaabeaa aOqaaaqaaiaaykW7caaMc8Uaey4kaSIaaGOmamaaqafabeWcbaGaam 4AaiabgIGiolaadohadaWgaaadbaGaaGymaaqabaWccaaMb8UaaGil aiaaysW7caWGSbGaeyicI4Saam4CamaaBaaameaacaaIZaaabeaaaS qab0GaeyyeIuoakiaadEhadaWgaaWcbaGaaGOmaiaadYgaaeqaaOGa am4DamaaBaaaleaacaaIZaGaamiBaaqabaGcdaqadaqaaiaadEhada qhaaWcbaGaaG4maiaadUgaaeaacaGGQaaaaOGaam4DamaaDaaaleaa caaIZaGaamiBaaqaaiaacQcaaaGccqGHsislcaWG3bWaa0baaSqaai aaiodacaWGRbGaamiBaaqaaiaacQcaaaaakiaawIcacaGLPaaaceWG LbGbaKaadaWgaaWcbaGaaGymaiaadUgaaeqaaOGabmyzayaajaWaaS baaSqaaiaaiodacaWGSbaabeaakiaaygW7caaIUaaaaaaa@FD7F@

e ^ 1 k = x 1 k γ ^ 1 x 2 k γ ^ 2 , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmyzayaaja WaaSbaaSqaaiaaigdacaWGRbaabeaakiaai2dacaWG4bWaa0baaSqa aiaaigdacaWGRbaabaGccWaGyBOmGikaaiqbeo7aNzaajaWaaSbaaS qaaiaaigdaaeqaaOGaeyOeI0IaamiEamaaDaaaleaacaaIYaGaam4A aaqaaOGamai2gkdiIcaacuaHZoWzgaqcamaaBaaaleaacaaIYaaabe aakiaaygW7caGGSaaaaa@4B93@ e ^ 2 k = x 2 k γ ^ 2 x 3 k γ ^ 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmyzayaaja WaaSbaaSqaaiaaikdacaWGRbaabeaakiaai2dacaWG4bWaa0baaSqa aiaaikdacaWGRbaabaGccWaGyBOmGikaaiqbeo7aNzaajaWaaSbaaS qaaiaaikdaaeqaaOGaeyOeI0IaamiEamaaDaaaleaacaaIZaGaam4A aaqaaOGamai2gkdiIcaacuaHZoWzgaqcamaaBaaaleaacaaIZaaabe aaaaa@4954@ et e ^ 3 k = x 3 k γ ^ 3 y k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfFv0dd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpe0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmyzayaaja WaaSbaaSqaaiaaiodacaWGRbaabeaakiaai2dacaWG4bWaa0baaSqa aiaaiodacaWGRbaabaGccWaGyBOmGikaaiqbeo7aNzaajaWaaSbaaS qaaiaaiodaaeqaaOGaeyOeI0IaamyEamaaBaaaleaacaWGRbaabeaa aaa@4310@ sont définis au théorème 3.1.

Bibliographie

Binder, D.A. (1996). Méthodes de linéarisation pour les échantillons à une et deux phases : Une approche de type « recette ». Techniques d’enquête, 22, 1, 17‑22. Article accessible à l’adresse http://www.statcan.gc.ca/pub/12-001-x/1996001/article/14389-fra.pdf.

Binder, D.A., Babyak, C., Brodeur, M., Hidiroglou, M. et Jocelyn, W. (2000). Variance estimation for two-phase stratified sampling. The Canadian Journal of Statistics, 28, 751-764.

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