Est-ce que la réduction du déséquilibre de la réponse accroît l’exactitude des estimations de l’enquête ? Section 7. Le premier résultat

Le résultat 1 s’applique au contexte d’enquête suivant : un échantillon autopondéré s MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpepC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4Caaaa@350F@ de taille n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpepC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaaaa@350A@ est tiré de U = { 1 , , k , , N } ; MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpepC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyvaiabg2 da9maacmaabaGaaGymaiaacYcacqWIMaYscaGGSaGaam4AaiaacYca cqWIMaYscaGGSaGaamOtaaGaay5Eaiaaw2haaiaacUdaaaa@4069@ le poids d k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpepC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamizamaaBa aaleaacaWGRbaabeaaaaa@361C@ est le même pour toutes les unités k . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpepC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4Aaiaac6 caaaa@35B9@ Le vecteur auxiliaire x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpepC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCiEaaaa@3518@ est un vecteur de groupes de dimension J , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpepC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOsaiaacY caaaa@3596@ de sorte que l’échantillon s MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpepC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4Caaaa@350F@ et l’ensemble de répondants r , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpepC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCaiaacY caaaa@35BE@ qui sont supposés être de tailles fixes m < n , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpepC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBaiabgY da8iaad6gacaGGSaaaaa@37B0@ sont divisés en J MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpepC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOsaaaa@34E6@ groupes non chevauchants. La notation pertinente est donnée à la fin de la section 3. Les valeurs y k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpepC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEamaaBa aaleaacaWGRbaabeaaaaa@3631@ sont traitées comme étant fixes, non aléatoires, comme cela se fait habituellement dans l’approche classique fondée sur le plan de sondage. Si les y k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpepC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEamaaBa aaleaacaWGRbaabeaaaaa@3631@ étaient observées pour tout k s , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpepC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4AaiabgI GiolaadohacaGGSaaaaa@3833@ alors Y ^ F U L = N y ¯ s , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpepC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmywayaaja WaaSbaaSqaaiaadAeacaWGvbGaamitaaqabaGccqGH9aqpcaWGobGa aGPaVlqadMhagaqeamaaBaaaleaacaWGZbaabeaakiaacYcaaaa@3E09@ avec y ¯ s = s y k / n , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpepC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmyEayaara WaaSbaaSqaaiaadohaaeqaaOGaeyypa0ZaaSGbaeaadaaeqaqaaiaa dMhadaWgaaWcbaGaam4AaaqabaaabaGaam4Caaqab0GaeyyeIuoaaO qaaiaad6gaaaGaaiilaaaa@3E0E@ serait sans biais sous le plan de sondage pour le total de population de y , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpepC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEaiaacY caaaa@35C5@ Y = U y k . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpepC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamywaiabg2 da9maaqababaGaamyEamaaBaaaleaacaWGRbaabeaaaeaacaWGvbaa beqdcqGHris5aOGaaiOlaaaa@3B83@ Mais y k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpepC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEamaaBa aaleaacaWGRbaabeaaaaa@3631@ est disponible pour k r MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpepC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4AaiabgI Giolaadkhaaaa@3782@ uniquement; l’estimateur CAL (5.1) devient Y ^ C A L = N j = 1 J W j s y ¯ r j , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpepC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmywayaaja WaaSbaaSqaaiaadoeacaWGbbGaamitaaqabaGccqGH9aqpcaWGobWa aabmaeaacaWGxbWaaSbaaSqaaiaadQgacaWGZbaabeaakiqadMhaga qeamaaBaaaleaacaWGYbWaaSbaaWqaaiaadQgaaeqaaaWcbeaaaeaa caWGQbGaeyypa0JaaGymaaqaaiaadQeaa0GaeyyeIuoakiaacYcaaa a@45FC@ y ¯ r j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpepC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmyEayaara WaaSbaaSqaaiaadkhadaWgaaadbaGaamOAaaqabaaaleqaaaaa@3777@ est la moyenne des valeurs y k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpepC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEamaaBa aaleaacaWGRbaabeaaaaa@3631@ des répondants dans le groupe j . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpepC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOAaiaac6 caaaa@35B8@ Les propriétés statistiques  MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqaqpepC0xbbL8F4rqqqpepeea0xe9Lqpe0x e9q8qqvqFr0dXdHiVc=bYP0xH8peuj0lXxdrpe0=1qpeeaY=rrVue9 Fve9Fve8meaabaqaciGacaGabeqabaWaaeaaeaaakeaaieaajugyba baaaaaaaaapeGaa83eGaaa@38B6@ l’espérance et la variance  MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqaqpepC0xbbL8F4rqqqpepeea0xe9Lqpe0x e9q8qqvqFr0dXdHiVc=bYP0xH8peuj0lXxdrpe0=1qpeeaY=rrVue9 Fve9Fve8meaabaqaciGacaGabeqabaWaaeaaeaaakeaaieaajugyba baaaaaaaaapeGaa83eGaaa@38B6@ de ( Y ^ C A L Y ^ F U L ) / N = Δ r MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpepC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSGbaeaada qadaqaaiqadMfagaqcamaaBaaaleaacaWGdbGaamyqaiaadYeaaeqa aOGaeyOeI0IabmywayaajaWaaSbaaSqaaiaadAeacaWGvbGaamitaa qabaaakiaawIcacaGLPaaaaeaacaWGobaaaiabg2da9iabfs5aenaa BaaaleaacaWGYbaabeaaaaa@4222@ avec Δ r = ( b r b s ) x ¯ s = j = 1 J W j s y ¯ r j y ¯ s MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpepC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuiLdq0aaS baaSqaaiaadkhaaeqaaOGaeyypa0ZaaeWaaeaacaWHIbWaaSbaaSqa aiaadkhaaeqaaOGaeyOeI0IaaCOyamaaBaaaleaacaWGZbaabeaaaO GaayjkaiaawMcaamaaCaaaleqabaGccWaGyBOmGikaaiqahIhagaqe amaaBaaaleaacaWGZbaabeaakiabg2da9maaqadabaGaam4vamaaBa aaleaacaWGQbGaam4CaaqabaGcceWG5bGbaebadaWgaaWcbaGaamOC amaaBaaameaacaWGQbaabeaaaSqabaGccqGHsislceWG5bGbaebada WgaaWcbaGaam4CaaqabaaabaGaamOAaiabg2da9iaaigdaaeaacaWG kbaaniabggHiLdaaaa@53BB@ sont données dans le résultat 1 pour les conditions probabilistes suivantes : tous les ( n   m   ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpepC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaada qhaaWcbaqcLbqacaWGTbaaleaajugabiaad6gaaaaakiaawIcacaGL Paaaaaa@38A4@ ensembles de réponses r MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpepC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCaaaa@350E@ de taille fixe m MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpepC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBaaaa@3509@ sont supposés a priori être également probables. Sachant s , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpepC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4CaiaacY caaaa@35BF@ le déséquilibre I M B MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpepC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysaiaad2 eacaWGcbaaaa@367E@ est déterminé par x ¯ r = ( 1 / m ) ( m 1 , , m j , , m J ) . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpepC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcaaMabCiEay aaraGcdaWgaaWcbaGaamOCaaqabaqcaaMaeyypa0Jcdaqadaqaamaa lyaabaGaaGymaaqaaiaad2gaaaaacaGLOaGaayzkaaWaaeWaaeaaja aycaWGTbGcdaWgaaqcbawaaiaaigdaaeqaaKaaGjaacYcakiablAci lLaaGjaacYcacaWGTbGcdaWgaaqcbawaaiaadQgaaeqaaKaaGjaacY cakiablAcilLaaGjaacYcacaWGTbGcdaWgaaqcbawaaiaadQeaaeqa aaGccaGLOaGaayzkaaWaaWbaaSqabeaakiadaITHYaIOaaqcaaMaai Olaaaa@4F83@ Sachant x ¯ r , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpepC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabCiEayaara WaaSbaaSqaaiaadkhaaeqaaOGaaiilaaaa@370D@ il nous reste R = j = 1 J ( n j m j ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpepC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOuaiabg2 da9maaradabaWaaeWaaeaajugabuaabeqaceaaaOqaaKqzaeGaamOB aKqbaoaaBaaaleaajugOaiaadQgaaSqabaaakeaajugabiaad2gaju aGdaWgaaWcbaqcLbkacaWGQbaaleqaaaaaaOGaayjkaiaawMcaaaWc baGaamOAaiabg2da9iaaigdaaeaacaWGkbaaniabg+Givdaaaa@45CD@ ensembles r , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpepC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCaiaacY caaaa@35BE@ ayant tous la même probabilité non nulle 1 / R MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpepC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSGbaeaaca aIXaaabaGaamOuaaaaaaa@35BF@ et le même I M B , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpepC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysaiaad2 eacaWGcbGaaiilaaaa@372E@ donné par (3.3). Les autres ensembles r MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpepC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCaaaa@350E@ de taille m MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpepC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBaaaa@3509@ ne sont plus dans le champ d’observation. Le conditionnement sur x ¯ r MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpepC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabCiEayaara WaaSbaaSqaaiaadkhaaeqaaaaa@3653@ nous permet d’étudier les propriétés de l’estimateur CAL en fonction de l’ I M B . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpepC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysaiaad2 eacaWGcbGaaiOlaaaa@3730@ Le résultat 1 fait intervenir la variance de la variable étudiée y , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpepC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEaiaacY caaaa@35C5@ à l’intérieur des groupes et combinée sur les groupes :

S y j 2 = s j ( y k y ¯ s j ) 2 / ( n j 1 ) ; S y 2 = j = 1 J W j s S y j 2 . ( 7.1 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpepC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcaaMaam4uaO Waa0baaKqaGfaacaWG5bGaamOAaaqaaiaaikdaaaqcaaMaeyypa0Jc daWcgaqaamaaqabajaaybaGcdaqadaqaaiaadMhadaWgaaWcbaGaam 4AaaqabaGccqGHsislceWG5bGbaebadaWgaaWcbaGaam4CamaaBaaa meaacaWGQbaabeaaaSqabaaakiaawIcacaGLPaaadaahaaqcbawabe aacaaIYaaaaaqaaiaadohalmaaBaaajiaybaGaamOAaaqabaaajeay beqcdaMaeyyeIuoaaOqaamaabmaabaqcaaMaamOBaOWaaSbaaKqaGf aacaWGQbaabeaajaaycqGHsislcaaIXaaakiaawIcacaGLPaaaaaqc aaMaai4oaOGaaGzbVlaadofadaqhaaWcbaGaamyEaaqaaiaaikdaaa GccqGH9aqpdaaeWbqaaiaadEfadaWgaaWcbaGaamOAaiaadohaaeqa aOGaam4uamaaDaaaleaacaWG5bGaamOAaaqaaiaaikdaaaaabaGaam OAaiabg2da9iaaigdaaeaacaWGkbaaniabggHiLdGccaGGUaGaaGzb VlaaywW7caaMf8UaaGzbVlaaywW7caGGOaGaaG4naiaac6cacaaIXa Gaaiykaaaa@6E88@

Résultat 1. Soit s MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpD0xh9Wqpm0db9Wq pepeuf0xe9q8qiYRWFGCk9vi=dbvc9s8vr0db9Fn0dbbG8Fq0Jfr=x fr=xfbpdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4Caaaa@3A50@ un échantillon autopondéré de taille n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpD0xh9Wqpm0db9Wq pepeuf0xe9q8qiYRWFGCk9vi=dbvc9s8vr0db9Fn0dbbG8Fq0Jfr=x fr=xfbpdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaaaa@3A4B@ et soit x k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpD0xh9Wqpm0db9Wq pepeuf0xe9q8qiYRWFGCk9vi=dbvc9s8vr0db9Fn0dbbG8Fq0Jfr=x fr=xfbpdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCiEamaaBa aaleaacaWGRbaabeaaaaa@3B75@ un vecteur de groupes de dimension  J . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpD0xh9Wqpm0db9Wq pepeuf0xe9q8qiYRWFGCk9vi=dbvc9s8vr0db9Fn0dbbG8Fq0Jfr=x fr=xfbpdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOsaiaac6 caaaa@3AD9@ Supposons que tous les ( n   m   ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpepC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaada qhaaWcbaqcLbqacaWGTbaaleaajugabiaad6gaaaaakiaawIcacaGL Paaaaaa@38A4@ ensembles de réponses r MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpD0xh9Wqpm0db9Wq pepeuf0xe9q8qiYRWFGCk9vi=dbvc9s8vr0db9Fn0dbbG8Fq0Jfr=x fr=xfbpdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCaaaa@3A4F@ de taille fixe m MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpD0xh9Wqpm0db9Wq pepeuf0xe9q8qiYRWFGCk9vi=dbvc9s8vr0db9Fn0dbbG8Fq0Jfr=x fr=xfbpdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBaaaa@3A4A@ sont a priori aussi probables les uns que les autres. Alors,

Δ ¯ = E ( Δ r | x ¯ r , m , s ) = 0 ( 7.2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpD0xh9Wqpm0db9Wq pepeuf0xe9q8qiYRWFGCk9vi=dbvc9s8vr0db9Fn0dbbG8Fq0Jfr=x fr=xfbpdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcaaQafuiLdq KbaebacqGH9aqpcaqGfbGcdaqadaqaamaaeiaabaqcaaQaeuiLdqKc daWgaaqcbaAaaiaadkhaaeqaaOGaaGPaVdGaayjcSdGaaGjbVlqahI hagaqeamaaBaaaleaacaWGYbaabeaakiaacYcacaWGTbGaaiilaiaa dohaaiaawIcacaGLPaaacqGH9aqpcaaIWaGaaGzbVlaaywW7caaMf8 UaaGzbVlaaywW7caGGOaGaaG4naiaac6cacaaIYaGaaiykaaaa@5A10@

S Δ 2 = E ( ( Δ r Δ ¯ ) 2 | x ¯ r , m , s ) = ( 1 m 1 n ) S y 2 + 1 m j = 1 J W j s ( p p j 1 ) S y j 2 ( 7.3 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpD0xh9Wqpm0db9Wq pepeuf0xe9q8qiYRWFGCk9vi=dbvc9s8vr0db9Fn0dbbG8Fq0Jfr=x fr=xfbpdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uamaaDa aaleaacqqHuoaraeaacaaIYaaaaOGaeyypa0JaaeyramaabmaabaWa aqGaaeaadaqadaqaaiabfs5aenaaBaaaleaacaWGYbaabeaakiabgk HiTiqbfs5aezaaraaacaGLOaGaayzkaaWaaWbaaSqabeaacaaIYaaa aOGaaGPaVdGaayjcSdGaaGjbVlqahIhagaqeamaaBaaaleaacaWGYb aabeaakiaacYcacaWGTbGaaiilaiaadohaaiaawIcacaGLPaaacqGH 9aqpdaqadaqaamaalaaabaGaaGymaaqaaiaad2gaaaGaeyOeI0YaaS aaaeaacaaIXaaabaGaamOBaaaaaiaawIcacaGLPaaacaWGtbWaa0ba aSqaaiaadMhaaeaacaaIYaaaaOGaey4kaSYaaSaaaeaacaaIXaaaba GaamyBaaaadaaeWbqaaiaadEfadaWgaaWcbaGaamOAaiaadohaaeqa aOWaaeWaaeaadaWcaaqaaiaadchaaeaacaWGWbWaaSbaaSqaaiaadQ gaaeqaaaaakiabgkHiTiaaigdaaiaawIcacaGLPaaacaWGtbWaa0ba aSqaaiaadMhacaWGQbaabaGaaGOmaaaaaeaacaWGQbGaeyypa0JaaG ymaaqaaiaadQeaa0GaeyyeIuoakiaaywW7caaMf8UaaGzbVlaaywW7 caaMf8UaaiikaiaaiEdacaGGUaGaaG4maiaacMcaaaa@7BFC@

W j s = n j / n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpD0xh9Wqpm0db9Wq pepeuf0xe9q8qiYRWFGCk9vi=dbvc9s8vr0db9Fn0dbbG8Fq0Jfr=x fr=xfbpdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4vamaaBa aaleaacaWGQbGaam4CaaqabaGccqGH9aqpdaWcgaqaaiaad6gadaWg aaWcbaGaamOAaaqabaaakeaacaWGUbaaaaaa@4078@ et p j = m j / n j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpD0xh9Wqpm0db9Wq pepeuf0xe9q8qiYRWFGCk9vi=dbvc9s8vr0db9Fn0dbbG8Fq0Jfr=x fr=xfbpdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCamaaBa aaleaacaWGQbaabeaakiabg2da9maalyaabaGaamyBamaaBaaaleaa caWGQbaabeaaaOqaaiaad6gadaWgaaWcbaGaamOAaaqabaaaaaaa@40B3@ sont la taille relative et le taux de réponse, respectivement, pour le groupe j , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpD0xh9Wqpm0db9Wq pepeuf0xe9q8qiYRWFGCk9vi=dbvc9s8vr0db9Fn0dbbG8Fq0Jfr=x fr=xfbpdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOAaiaacY caaaa@3AF7@ p = m / n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpD0xh9Wqpm0db9Wq pepeuf0xe9q8qiYRWFGCk9vi=dbvc9s8vr0db9Fn0dbbG8Fq0Jfr=x fr=xfbpdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCaiabg2 da9maalyaabaGaamyBaaqaaiaad6gaaaaaaa@3D4E@ est le taux de réponse global, et S y 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpD0xh9Wqpm0db9Wq pepeuf0xe9q8qiYRWFGCk9vi=dbvc9s8vr0db9Fn0dbbG8Fq0Jfr=x fr=xfbpdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uamaaDa aaleaacaWG5baabaGaaGOmaaaaaaa@3C17@ et S y j 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpD0xh9Wqpm0db9Wq pepeuf0xe9q8qiYRWFGCk9vi=dbvc9s8vr0db9Fn0dbbG8Fq0Jfr=x fr=xfbpdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uamaaDa aaleaacaWG5bGaamOAaaqaaiaaikdaaaaaaa@3D06@ sont données en (7.1). Si les taux de réponse p j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpD0xh9Wqpm0db9Wq pepeuf0xe9q8qiYRWFGCk9vi=dbvc9s8vr0db9Fn0dbbG8Fq0Jfr=x fr=xfbpdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCamaaBa aaleaacaWGQbaabeaaaaa@3B68@ et les variances S y j 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpD0xh9Wqpm0db9Wq pepeuf0xe9q8qiYRWFGCk9vi=dbvc9s8vr0db9Fn0dbbG8Fq0Jfr=x fr=xfbpdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uamaaDa aaleaacaWG5bGaamOAaaqaaiaaikdaaaaaaa@3D06@ ne varient que peu sur les groupes, alors

S Δ 2 ( 1 p + I M B p 2 ) S y 2 m ( 7.4 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpD0xh9Wqpm0db9Wq pepeuf0xe9q8qiYRWFGCk9vi=dbvc9s8vr0db9Fn0dbbG8Fq0Jfr=x fr=xfbpdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uamaaDa aaleaacqqHuoaraeaacaaIYaaaaOGaeyisIS7aaeWaaeaacaaIXaGa eyOeI0IaamiCaKqzGhGaey4kaSIcdaWcaaqaaiaadMeacaWGnbGaam OqaaqaaiaadchadaahaaWcbeqaaiaaikdaaaaaaaGccaGLOaGaayzk aaWaaSaaaeaacaWGtbWaa0baaSqaaiaadMhaaeaacaaIYaaaaaGcba GaamyBaaaacaaMf8UaaGzbVlaaywW7caaMf8UaaGzbVlaacIcacaaI 3aGaaiOlaiaaisdacaGGPaaaaa@5814@

I M B MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpD0xh9Wqpm0db9Wq pepeuf0xe9q8qiYRWFGCk9vi=dbvc9s8vr0db9Fn0dbbG8Fq0Jfr=x fr=xfbpdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysaiaad2 eacaWGcbaaaa@3BBF@ est donné par (3.3).

En cas de réponse complète, quand r = s , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpD0xh9Wqpm0db9Wq pepeuf0xe9q8qiYRWFGCk9vi=dbvc9s8vr0db9Fn0dbbG8Fq0Jfr=x fr=xfbpdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCaiabg2 da9iaadohacaGGSaaaaa@3CFD@ les deuxièmes membres des équations (7.3) et (7.4) sont nuls; l’approximation dans (7.4) est exacte : S Δ 2 = 0. MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpD0xh9Wqpm0db9Wq pepeuf0xe9q8qiYRWFGCk9vi=dbvc9s8vr0db9Fn0dbbG8Fq0Jfr=x fr=xfbpdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uamaaDa aaleaacqqHuoaraeaacaaIYaaaaOGaeyypa0JaaGimaiaac6caaaa@3EFB@ Pour interpréter le résultat 1, notons que le premier terme du deuxième membre de (7.3) est une constante, sachant m . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpD0xh9Wqpm0db9Wq pepeuf0xe9q8qiYRWFGCk9vi=dbvc9s8vr0db9Fn0dbbG8Fq0Jfr=x fr=xfbpdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBaiaac6 caaaa@3AFC@ Ce terme donne la variance conditionnelle pour une réponse parfaitement équilibrée, où p j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpD0xh9Wqpm0db9Wq pepeuf0xe9q8qiYRWFGCk9vi=dbvc9s8vr0db9Fn0dbbG8Fq0Jfr=x fr=xfbpdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCamaaBa aaleaacaWGQbaabeaaaaa@3B68@ est la même pour tous les groupes. Le deuxième terme est le terme de pénalisation, à savoir la pénalisation du fait de ne pas obtenir l’équilibre parfait durant la collecte des données. La taille de ce terme dépend de la mesure dans laquelle le plan de collecte dynamique réussit à générer des taux de réponse de groupe p j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpD0xh9Wqpm0db9Wq pepeuf0xe9q8qiYRWFGCk9vi=dbvc9s8vr0db9Fn0dbbG8Fq0Jfr=x fr=xfbpdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcaaMaamiCaO WaaSbaaKqaGfaacaWGQbaabeaaaaa@3C3A@ qui ne varient que faiblement. Ce terme est nul si l’on peut rendre tous les p j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpD0xh9Wqpm0db9Wq pepeuf0xe9q8qiYRWFGCk9vi=dbvc9s8vr0db9Fn0dbbG8Fq0Jfr=x fr=xfbpdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcaaMaamiCaO WaaSbaaKqaGfaacaWGQbaabeaaaaa@3C3A@ égaux.

La formule (7.4) indique que la variance S Δ 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpD0xh9Wqpm0db9Wq pepeuf0xe9q8qiYRWFGCk9vi=dbvc9s8vr0db9Fn0dbbG8Fq0Jfr=x fr=xfbpdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uamaaDa aaleaacqqHuoaraeaacaaIYaaaaaaa@3C7F@ diminue parallèlement à l’ I M B MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpD0xh9Wqpm0db9Wq pepeuf0xe9q8qiYRWFGCk9vi=dbvc9s8vr0db9Fn0dbbG8Fq0Jfr=x fr=xfbpdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysaiaad2 eacaWGcbaaaa@3BBF@ de manière plus ou moins linéaire. Donc, un faible déséquilibre accroît les chances d’obtenir un petit écart Y ^ C A L Y ^ F U L = N ^ Δ r , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpD0xh9Wqpm0db9Wq pepeuf0xe9q8qiYRWFGCk9vi=dbvc9s8vr0db9Fn0dbbG8Fq0Jfr=x fr=xfbpdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmywayaaja WaaSbaaSqaaiaadoeacaWGbbGaamitaaqabaGccqGHsislceWGzbGb aKaadaWgaaWcbaGaamOraiaadwfacaWGmbaabeaakiabg2da9iqad6 eagaqcaiabfs5aenaaBaaaleaacaWGYbaabeaakiaacYcacaaMe8oa aa@481B@ ce qui est important en pratique. Par exemple, pour une non-réponse de 1 p = 40 % , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpD0xh9Wqpm0db9Wq pepeuf0xe9q8qiYRWFGCk9vi=dbvc9s8vr0db9Fn0dbbG8Fq0Jfr=x fr=xfbpdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiabgk HiTiaadchacqGH9aqpcaaI0aGaaGimaiaaysW7caGGLaGaaiilaaaa @4159@ S Δ 2 0 , 57 S y 2 / m MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpD0xh9Wqpm0db9Wq pepeuf0xe9q8qiYRWFGCk9vi=dbvc9s8vr0db9Fn0dbbG8Fq0Jfr=x fr=xfbpdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uamaaDa aaleaacqqHuoaraeaacaaIYaaaaOGaeyisIS7aaSGbaeaacaaIWaGa aiilaiaaiwdacaaI3aGaam4uamaaDaaaleaacaWG5baabaGaaGOmaa aaaOqaaiaad2gaaaaaaa@44F5@ si I M B = 0 , 06 , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpD0xh9Wqpm0db9Wq pepeuf0xe9q8qiYRWFGCk9vi=dbvc9s8vr0db9Fn0dbbG8Fq0Jfr=x fr=xfbpdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysaiaad2 eacaWGcbGaeyypa0JaaGimaiaacYcacaaIWaGaaGOnaiaacYcaaaa@4059@ mais si I M B = 0 , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpD0xh9Wqpm0db9Wq pepeuf0xe9q8qiYRWFGCk9vi=dbvc9s8vr0db9Fn0dbbG8Fq0Jfr=x fr=xfbpdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysaiaad2 eacaWGcbGaeyypa0JaaGimaiaacYcaaaa@3E2F@ c’est-à-dire en cas d’équilibre parfait, cette variance est réduite à S Δ 2 0 , 40 S y 2 / m . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpD0xh9Wqpm0db9Wq pepeuf0xe9q8qiYRWFGCk9vi=dbvc9s8vr0db9Fn0dbbG8Fq0Jfr=x fr=xfbpdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uamaaDa aaleaacqqHuoaraeaacaaIYaaaaOGaeyisIS7aaSGbaeaacaaIWaGa aiilaiaaisdacaaIWaGaam4uamaaDaaaleaacaWG5baabaGaaGOmaa aaaOqaaiaad2gaaaGaaiOlaaaa@459F@ L’amélioration est claire, mais ne peut être qualifiée de très grande. Cela tient au fait que I M B / p 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpD0xh9Wqpm0db9Wq pepeuf0xe9q8qiYRWFGCk9vi=dbvc9s8vr0db9Fn0dbbG8Fq0Jfr=x fr=xfbpdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSGbaeaaca WGjbGaamytaiaadkeaaeaacaWGWbWaaWbaaSqabeaacaaIYaaaaaaa aaa@3DB3@ est souvent faible comparativement à une non-réponse 1 p MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpD0xh9Wqpm0db9Wq pepeuf0xe9q8qiYRWFGCk9vi=dbvc9s8vr0db9Fn0dbbG8Fq0Jfr=x fr=xfbpdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiabgk HiTiaadchaaaa@3BF5@ de l’ordre de 30 à 60 %, c’est-à-dire les cas dont nous nous préoccupons principalement ici. Donc, prendre des mesures en vue de réduire le déséquilibre a un effet désirable, mais modeste plutôt que puissant.

Dans (7.2) et (7.3), l’espérance E ( ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpD0xh9Wqpm0db9Wq pepeuf0xe9q8qiYRWFGCk9vi=dbvc9s8vr0db9Fn0dbbG8Fq0Jfr=x fr=xfbpdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeyramaabm aabaGaeyyXICnacaGLOaGaayzkaaaaaa@3DF3@ est obtenue en prenant la moyenne sur les R = j = 1 J ( n j m j ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpD0xh9Wqpm0db9Wq pepeuf0xe9q8qiYRWFGCk9vi=dbvc9s8vr0db9Fn0dbbG8Fq0Jfr=x fr=xfbpdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOuaiabg2 da9maaradabaWaaeWaaeaafaqabeGabaaabaqcLbqacaWGUbGcdaWg aaWcbaqcLbkacaWGQbaaleqaaaGcbaqcLbqacaWGTbGcdaWgaaWcba qcLbkacaWGQbaaleqaaaaaaOGaayjkaiaawMcaaaWcbaGaamOAaiab g2da9iaaigdaaeaacaWGkbaaniabg+Givdaaaa@498E@ ensembles équiprobables r MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpD0xh9Wqpm0db9Wq pepeuf0xe9q8qiYRWFGCk9vi=dbvc9s8vr0db9Fn0dbbG8Fq0Jfr=x fr=xfbpdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCaaaa@3A4F@ qui restent parmi ( n   m   ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpepC0xd9Wqpe0dd9 qqaqFeFr0xbbG8FaYPYRWFb9fi0lXxbvc9Ff0dfrpm0dXdHqps0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaada qhaaWcbaqcLbqacaWGTbaaleaajugabiaad6gaaaaakiaawIcacaGL Paaaaaa@38A4@ après avoir fixé x ¯ r . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpD0xh9Wqpm0db9Wq pepeuf0xe9q8qiYRWFGCk9vi=dbvc9s8vr0db9Fn0dbbG8Fq0Jfr=x fr=xfbpdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabCiEayaara WaaSbaaSqaaiaadkhaaeqaaOGaaiOlaaaa@3C50@ Il convient aussi de souligner que plus d’une moyenne x ¯ r MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpD0xh9Wqpm0db9Wq pepeuf0xe9q8qiYRWFGCk9vi=dbvc9s8vr0db9Fn0dbbG8Fq0Jfr=x fr=xfbpdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabCiEayaara WaaSbaaSqaaiaadkhaaeqaaaaa@3B94@ peut donner la même valeur I M B . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpD0xh9Wqpm0db9Wq pepeuf0xe9q8qiYRWFGCk9vi=dbvc9s8vr0db9Fn0dbbG8Fq0Jfr=x fr=xfbpdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysaiaad2 eacaWGcbGaaiOlaaaa@3C71@ Donc, il peut exister plus d’une valeur S Δ 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpD0xh9Wqpm0db9Wq pepeuf0xe9q8qiYRWFGCk9vi=dbvc9s8vr0db9Fn0dbbG8Fq0Jfr=x fr=xfbpdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uamaaDa aaleaacqqHuoaraeaacaaIYaaaaaaa@3C7F@ pour le même I M B . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpD0xh9Wqpm0db9Wq pepeuf0xe9q8qiYRWFGCk9vi=dbvc9s8vr0db9Fn0dbbG8Fq0Jfr=x fr=xfbpdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysaiaad2 eacaWGcbGaaiOlaaaa@3C71@ La fonction linéairement croissante de I M B MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpD0xh9Wqpm0db9Wq pepeuf0xe9q8qiYRWFGCk9vi=dbvc9s8vr0db9Fn0dbbG8Fq0Jfr=x fr=xfbpdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysaiaad2 eacaWGcbaaaa@3BBF@ en (7.4) est néanmoins leur approximation commune.

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