2 Estimation bayésienne linéaire en population finie

Kelly Cristina M. Gonçalves, Fernando A. S. Moura et Helio S. Migon

Précédent | Suivant

L'approche bayésienne s'est avérée fructueuse dans de nombreuses applications, particulièrement lorsque l'analyse des données a été améliorée par des jugements d'expert. Cependant, si les modèles bayésiens possèdent de nombreuses caractéristiques intéressantes, leur application requiert souvent la spécification complète d'une loi a priori, ou prior, pour un grand nombre de paramètres. Goldstein et Wooff (2007), section 1.2, soutiennent que, à mesure que le problème se complexifie, notre aptitude réelle à spécifier complètement la loi a priori et/ou le modèle d'échantillonnage en détail diminue. Ils concluent que, dans de telles situations, il est nécessaire d'élaborer des méthodes qui reposent sur une spécification faisant appel à la croyance partielle.

Hartigan (1969) a proposé une méthode d'estimation, qu'il a nommée approche d'estimation bayésienne linéaire, qui ne nécessite que la spécification des premier et deuxième moments. Les estimateurs résultants ont la propriété de minimiser la perte quadratique a posteriori parmi tous les estimateurs qui sont linéaires en les données et peuvent être considérés comme des approximations des moyennes a posteriori. L'approche d'estimation bayésienne linéaire, qui est employée pleinement dans le présent article, est décrite brièvement ci-dessous.

2.1  Approche bayésienne linéaire

Soit y s MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai aahMhadaWgaaWcbaGaam4Caaqabaaaaa@3D3D@  le vecteur des observations et θ, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai aahI7acaGGSaaaaa@3D0B@  le paramètre à estimer. Pour chaque valeur de θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai aahI7aaaa@3C5B@  et chaque estimation possible d, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai aahsgacaGGSaaaaa@3CB4@  appartenant à l'espace paramétrique Θ, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai aahI5acaGGSaaaaa@3CEB@  nous associons une fonction de perte quadratique L( θ,d )=( θd )( θd )=tr( θd )( θd ). MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai aadYeadaqadaqaaiaahI7acaaISaGaaCizaaGaayjkaiaawMcaaiab g2da9maabmaabaGaaCiUdiabgkHiTiaahsgaaiaawIcacaGLPaaaii aacqWFYaIOdaqadaqaaiaahI7acqGHsislcaWHKbaacaGLOaGaayzk aaGaeyypa0JaamiDaiaadkhadaqadaqaaiaahI7acqGHsislcaWHKb aacaGLOaGaayzkaaWaaeWaaeaacaWH4oGaeyOeI0IaaCizaaGaayjk aiaawMcaaiab=jdiIkab=5caUaaa@5ACD@  Nous souhaitons avant tout trouver la valeur de d MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai aahsgaaaa@3C04@  qui minimise r( d )=E[ L( θ,d )| y s ], MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai aadkhadaqadaqaaiaahsgaaiaawIcacaGLPaaacqGH9aqpcaWGfbWa amWaaeaadaabcaqaaiaadYeadaqadaqaaiaahI7acaaISaGaaCizaa GaayjkaiaawMcaaiaaykW7aiaawIa7aiaahMhadaWgaaWcbaGaam4C aaqabaaakiaawUfacaGLDbaacaGGSaaaaa@4D88@  la valeur prévue conditionnelle de la fonction de perte quadratique fournie par les données.

Supposons que la distribution conjointe de θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai aahI7aaaa@3C5B@  et y s MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai aahMhadaWgaaWcbaGaam4Caaqabaaaaa@3D3D@  est spécifiée partiellement par leurs deux premiers moments seulement :

( θ y s )[ ( a f ),( R AQ Q A Q ) ],          (2.1) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaam aabmaabaqbaeqabiqaaaqaaiaahI7aaeaacaWH5bWaaSbaaSqaaiaa dohaaeqaaaaaaOGaayjkaiaawMcaaGabaiab=XJi6maadmaabaWaae WaaeaafaqabeGabaaabaGaaCyyaaqaaiaahAgaaaaacaGLOaGaayzk aaGaaGilamaabmaabaqbaeqabiGaaaqaaiaahkfaaeaacaWHbbGaaC yuaaqaaiaahgfaceWHbbGbauaaaeaacaWHrbaaaaGaayjkaiaawMca aaGaay5waiaaw2faaiaaiYcacaqGGaGaaeiiaiaabccacaqGGaGaae iiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeikaiaabkdacaqG UaGaaeymaiaabMcaaaa@5888@

a MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai aahggaaaa@3C01@  et f, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai aahAgacaGGSaaaaa@3CB6@  respectivement, désignent les vecteurs des moyennes et R, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai aahkfacaGGSaaaaa@3CA2@   AQ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai aahgeacaWHrbaaaa@3CBB@  et Q, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai aahgfacaGGSaaaaa@3CA1@  les éléments de la matrice de covariance de θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai aahI7aaaa@3C5B@  et y s . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai aahMhadaWgaaWcbaGaam4CaaqabaGccaaIUaaaaa@3DFF@

L'estimateur bayésien linéaire (EBL) de θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai aahI7aaaa@3C5B@  est la valeur de d MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai aahsgaaaa@3C04@  qui minimise la valeur prévue de cette fonction de perte quadratique dans la classe de toutes les estimations linéaires de la forme d=d( y s )=h+H y s , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai aahsgacqGH9aqpcaWHKbWaaeWaaeaacaWH5bWaaSbaaSqaaiaadoha aeqaaaGccaGLOaGaayzkaaGaeyypa0JaaCiAaiabgUcaRiaahIeaca WH5bWaaSbaaSqaaiaadohaaeqaaOGaaiilaaaa@483A@  pour un vecteur h MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai aahIgaaaa@3C08@  et une matrice H. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai aahIeacaGGUaaaaa@3C9A@  Donc, l'EBL de θ, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai aahI7acaGGSaaaaa@3D0B@   d ^ , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai qahsgagaqcaiaacYcaaaa@3CC4@  et sa variance associée, V ^ ( d ^ ), MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai qadAfagaqcamaabmaabaGabCizayaajaaacaGLOaGaayzkaaGaaiil aaaa@3F38@  sont donnés respectivement par

d ^ =a+A( y s f )et V ^ ( d ^ )=RAQA.         (2.2) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai qahsgagaqcaiabg2da9iaahggacqGHRaWkcaWHbbWaaeWaaeaacaWH 5bWaaSbaaSqaaiaadohaaeqaaOGaeyOeI0IaaCOzaaGaayjkaiaawM caaiaaykW7caaMc8UaaeyzaiaabshacaaMc8UaaGPaVlqadAfagaqc amaabmaabaGabCizayaajaaacaGLOaGaayzkaaGaeyypa0JaaCOuai abgkHiTiaahgeacaWHrbGaaCyqaGGaaiab=jdiIkaai6cacaqGGaGa aeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccaca qGOaGaaeOmaiaab6cacaqGYaGaaeykaaaa@6165@

Il convient de souligner que l'EBL dépend de la spécification des premier et deuxième moments de la distribution conjointe partiellement spécifiée en (2.1). Le problème de l'obtention de ces quantités est traité aux sections 2.3.1 et 4.1 pour certains cas particuliers.

2.2  Approche bayésienne linéaire en population finie

Considérons U={ u 1 ,, u N } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai aadwfacqGH9aqpdaGadaqaaiaadwhadaWgaaWcbaGaaGymaaqabaGc caaISaGaeSOjGSKaaGilaiaadwhadaWgaaWcbaGaamOtaaqabaaaki aawUhacaGL9baaaaa@45A4@  une population finie comprenant N MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai aad6eaaaa@3BEA@  unités. Soit y=( y 1 ,, y N ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai aahMhacqGH9aqpdaqadaqaaiaadMhadaWgaaWcbaGaaGymaaqabaGc caaISaGaeSOjGSKaaGilaiaadMhadaWgaaWcbaGaamOtaaqabaaaki aawIcacaGLPaaaiiaacqWFYaIOaaa@46AF@  le vecteur des valeurs d'intérêt des unités dans U. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai aadwfacaGGUaaaaa@3CA3@  Le vecteur de réponses y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai aahMhaaaa@3C19@  est divisé en le vecteur des valeurs connues observées sur l'échantillon y s MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai aahMhadaWgaaWcbaGaam4Caaqabaaaaa@3D3D@  de taille n, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai aad6gacaGGSaaaaa@3CBA@  et en le vecteur des valeurs non observées y s ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai aahMhadaWgaaWcbaGabm4Cayaaraaabeaaaaa@3D55@  de dimension Nn. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai aad6eacqGHsislcaWGUbGaaiOlaaaa@3E7C@  Le problème général consiste à prédire une fonction du vecteur y, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai aahMhacaGGSaaaaa@3CC9@  telle que le total  T= i=1 N y i = 1 s y s + 1 s ¯ y s ¯ , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai aadsfacqGH9aqpdaaeWaqabSqaaiaadMgacqGH9aqpcaaIXaaabaGa amOtaaqdcqGHris5aOGaaGPaVlaadMhadaWgaaWcbaGaamyAaaqaba GccqGH9aqpceWHXaGbauaadaWgaaWcbaGaam4CaaqabaGccaWH5bWa aSbaaSqaaiaadohaaeqaaOGaey4kaSIabCymayaafaWaaSbaaSqaai qadohagaqeaaqabaGccaWH5bWaaSbaaSqaaiqadohagaqeaaqabaGc caGGSaaaaa@5142@  où 1 s MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai aahgdadaWgaaWcbaGaam4Caaqabaaaaa@3CF5@  et 1 s ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai aahgdadaWgaaWcbaGabm4Cayaaraaabeaaaaa@3D0D@  sont les vecteurs de 1 de dimensions n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai aad6gaaaa@3C0A@  et Nn, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai aad6eacqGHsislcaWGUbGaaiilaaaa@3E7A@  respectivement. Dans l'approche fondée sur un modèle, cela se fait habituellement en se servant d'un modèle paramétrique hypothétique pour les valeurs de population y i , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai aadMhadaWgaaWcbaGaamyAaaqabaacbaGccaWFSaaaaa@3DEF@  puis en obtenant le meilleur prédicteur linéaire sans biais empirique (EBLUP) pour le vecteur inconnu y s ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai aahMhadaWgaaWcbaGabm4Cayaaraaabeaaaaa@3D55@  sous ce modèle. Habituellement, l'erreur quadratique moyenne de l'EBLUP de T MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai aadsfaaaa@3BF0@  s'obtient par approximation d'ordre deux, de même qu'un estimateur sans biais. Voir Valliant, Dorfman et Royall (2000), chapitre 2, pour des renseignements détaillés.

L'approche bayésienne de la prédiction en population finie repose souvent sur l'hypothèse d'un modèle paramétrique, mais elle vise à trouver la loi a posteriori de T MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai aadsfaaaa@3BF0@  sachant y s . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai aahMhadaWgaaWcbaGaam4CaaqabaGccaGGUaaaaa@3DF9@  On peut obtenir les estimations ponctuelles en spécifiant une fonction de perte, quoique dans de nombreux problèmes pratiques, on considère souvent la moyenne a posteriori dont la variance associée est donnée par la variance a posteriori, c'est-à-dire :

E( T| y s )= 1 s y s + 1 s ¯ E( y s ¯ | y s )etV( T| y s )= 1 s ¯ V( y s ¯ | y s ) 1 s ¯ .         (2.3) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai aadweadaqadaqaamaaeiaabaGaamivaiaaykW7aiaawIa7aiaahMha daWgaaWcbaGaam4CaaqabaaakiaawIcacaGLPaaacqGH9aqpceWHXa GbauaadaWgaaWcbaGaam4CaaqabaGccaWH5bWaaSbaaSqaaiaadoha aeqaaOGaey4kaSIabCymayaafaWaaSbaaSqaaiqadohagaqeaaqaba GccaWGfbWaaeWaaeaadaabcaqaaiaahMhadaWgaaWcbaGabm4Cayaa raaabeaakiaaykW7aiaawIa7aiaahMhadaWgaaWcbaGaam4Caaqaba aakiaawIcacaGLPaaacaaMc8UaaGPaVlaabwgacaqG0bGaaGPaVlaa ykW7caWGwbWaaeWaaeaadaabcaqaaiaadsfacaaMc8oacaGLiWoaca WH5bWaaSbaaSqaaiaadohaaeqaaaGccaGLOaGaayzkaaGaeyypa0Ja bCymayaafaWaaSbaaSqaaiqadohagaqeaaqabaGccaWGwbWaaeWaae aadaabcaqaaiaahMhadaWgaaWcbaGabm4CayaaraaabeaakiaaykW7 aiaawIa7aiaahMhadaWgaaWcbaGaam4CaaqabaaakiaawIcacaGLPa aacaWHXaWaaSbaaSqaaiqadohagaqeaaqabaGccaaIUaGaaeiiaiaa bccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaae ikaiaabkdacaqGUaGaae4maiaabMcaaaa@7F26@

Il est possible d'obtenir une approximation des quantités dans (2.3) en utilisant une approche d'estimation bayésienne linéaire. Ici, nous obtiendrons en particulier les estimateurs en émettant l'hypothèse d'un modèle hiérarchique à deux degrés général en population finie, spécifié uniquement par sa moyenne et sa matrice de variance-covariance, présenté dans Bolfarine et Zacks (1992), page 76. Les cas particuliers décrivant les structures de population habituellement observées en pratique peuvent être dérivés facilement de (2.4). Le modèle général peut s'écrire :

y|β[ Xβ,V ]etβ[ a,R ],         (2.4) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaam aaeiaabaGaaCyEaiaaykW7aiaawIa7aiaahk7aiqaacqWF8iIodaWa daqaaiaahIfacaWHYoGaaeilaiaahAfaaiaawUfacaGLDbaacaaMc8 UaaGPaVlaabwgacaqG0bGaaGPaVlaaykW7caWHYoGae8hpIOZaamWa aeaacaWHHbGaaGilaiaahkfaaiaawUfacaGLDbaacaaISaGaaeiiai aabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGa aeikaiaaikdacaGGUaGaaGinaiaacMcaaaa@6008@

X MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai aahIfaaaa@3BF8@  est une matrice de covariables de dimensions N×p, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai aad6eacqGHxdaTcaWGWbGaaiilaaaa@3FA6@  avec les lignes X i =( x i1 ,, x ip ), MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai aahIfadaWgaaWcbaGaamyAaaqabaGccqGH9aqpdaqadaqaaiaadIha daWgaaWcbaGaamyAaiaaigdaaeqaaOGaaGilaiablAciljaaiYcaca WG4bWaaSbaaSqaaiaadMgacaWGWbaabeaaaOGaayjkaiaawMcaaiaa cYcaaaa@48DB@   i=1,,N; MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai aadMgacqGH9aqpcaaIXaGaaGilaiablAciljaaiYcacaWGobGaai4o aaaa@41E6@   β= ( β 1 ,, β p ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai aahk7acqGH9aqpdaqadaqaaiabek7aInaaBaaaleaacaaIXaaabeaa kiaaiYcacqWIMaYscaaISaGaeqOSdi2aaSbaaSqaaiaadchaaeqaaa GccaGLOaGaayzkaaWaaWbaaSqabeaakiadacUHYaIOaaaaaa@49EF@  est un vecteur de dimension p×1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai aadchacqGHxdaTcaaIXaaaaa@3EDE@  de paramètres inconnus, et y, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai aahMhacaGGSaaaaa@3CC9@  sachant β, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai aahk7acaGGSaaaaa@3D05@  est un vecteur aléatoire de moyenne Xβ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai aahIfacaWHYoaaaa@3D36@  et de matrice de covariance connue V MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai aahAfaaaa@3BF6@  de dimensions N×N. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai aad6eacqGHxdaTcaWGobGaaiOlaaaa@3F86@  De manière analogue, a MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai aahggaaaa@3C01@  et R MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai aahkfaaaa@3BF2@  sont, respectivement, le vecteur des moyennes a priori de dimension p×1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai aadchacqGHxdaTcaaIXaaaaa@3EDE@  et la matrice de covariance a priori de dimensions p×p MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai aadchacqGHxdaTcaWGWbaaaa@3F18@  de β. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai aahk7acaGGUaaaaa@3D07@

Puisque le vecteur de réponses y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai aahMhaaaa@3C19@  est divisé en y s MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai aahMhadaWgaaWcbaGaam4Caaqabaaaaa@3D3D@  et y s ¯ , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai aahMhadaWgaaWcbaGabm4CayaaraaabeaakiaacYcaaaa@3E0F@  la matrice X, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai aahIfacaGGSaaaaa@3CA8@  qui est supposée connue, est divisée de manière analogue en X s MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai aahIfadaWgaaWcbaGaam4Caaqabaaaaa@3D1C@  et X s ¯ , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai aahIfadaWgaaWcbaGabm4CayaaraaabeaakiaacYcaaaa@3DEE@  et V MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai aahAfaaaa@3BF7@  est divisée en V s , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai aahAfadaWgaaWcbaGaam4CaaqabaGccaGGSaaaaa@3DD4@   V s ¯ , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai aahAfadaWgaaWcbaGabm4CayaaraaabeaakiaacYcaaaa@3DEC@   V s s ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai aahAfadaWgaaWcbaGaam4Caiqadohagaqeaaqabaaaaa@3E2A@  et V s ¯ s . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai aahAfadaWgaaWcbaGabm4CayaaraGaam4CaaqabaGccaGGUaaaaa@3EE6@  L'objectif premier est de prédire y s ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai aahMhadaWgaaWcbaGabm4Cayaaraaabeaaaaa@3D55@  sachant l'échantillon observé y s , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai aahMhadaWgaaWcbaGaam4CaaqabaGccaGGSaaaaa@3DF7@  puis le total T. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai aadsfacaGGUaaaaa@3CA2@  Voici les étapes que nous avons suivies : premièrement, nous avons utilisé une loi a priori conjointe qui n'est spécifiée que partiellement en ce qui concerne les moments comme il suit :

( y s ¯ y s )|β[ ( X s ¯ β X s β ),( V s ¯ V s ¯ s V s s ¯ V s ) ]. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaam aaeiaabaWaaeWaaeaafaqabeGabaaabaGaaCyEamaaBaaaleaaceWG ZbGbaebaaeqaaaGcbaGaaCyEamaaBaaaleaacaWGZbaabeaaaaaaki aawIcacaGLPaaacaaMc8oacaGLiWoacaWHYoaceaGae8hpIOZaamWa aeaadaqadaqaauaabeqaceaaaeaacaWHybWaaSbaaSqaaiqadohaga qeaaqabaGccaWHYoaabaGaaCiwamaaBaaaleaacaWGZbaabeaakiaa hk7aaaaacaGLOaGaayzkaaGaaGilamaabmaabaqbaeqabiGaaaqaai aahAfadaWgaaWcbaGabm4CayaaraaabeaaaOqaaiaahAfadaWgaaWc baGabm4CayaaraGaam4CaaqabaaakeaacaWHwbWaaSbaaSqaaiaado haceWGZbGbaebaaeqaaaGcbaGaaCOvamaaBaaaleaacaWGZbaabeaa aaaakiaawIcacaGLPaaaaiaawUfacaGLDbaacaaIUaaaaa@5E5A@

Donc, en appliquant le résultat général dans l'équation (2.2), l'EBL de E( y s ¯ | y s ,β ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai aadweadaqadaqaamaaeiaabaGaaCyEamaaBaaaleaaceWGZbGbaeba aeqaaaGccaGLiWoacaWH5bWaaSbaaSqaaiaadohaaeqaaOGaaGilai aahk7aaiaawIcacaGLPaaaaaa@456C@  et la perte quadratique prévue minimale (variance associée) sont donnés par :

E ^ ( y s ¯ | y s ,β )= X s ¯ β+ V s ¯ s V s 1 ( y s X s β )et V ^ ( y s ¯ | y s ,β )= V s ¯ V s ¯ s V s 1 V s s ¯ .         (2.5) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai qadweagaqcamaabmaabaWaaqGaaeaacaWH5bWaaSbaaSqaaiqadoha gaqeaaqabaaakiaawIa7aiaahMhadaWgaaWcbaGaam4CaaqabaGcca aISaGaaCOSdaGaayjkaiaawMcaaiabg2da9iaahIfadaWgaaWcbaGa bm4Cayaaraaabeaakiaahk7acqGHRaWkcaWHwbWaaSbaaSqaaiqado hagaqeaiaadohaaeqaaOGaaCOvamaaDaaaleaacaWGZbaabaGaeyOe I0IaaGymaaaakmaabmaabaGaaCyEamaaBaaaleaacaWGZbaabeaaki abgkHiTiaahIfadaWgaaWcbaGaam4CaaqabaGccaWHYoaacaGLOaGa ayzkaaGaaGPaVlaaykW7caqGLbGaaeiDaiaaykW7caaMc8UabmOvay aajaWaaeWaaeaadaabcaqaaiaahMhadaWgaaWcbaGabm4Cayaaraaa beaaaOGaayjcSdGaaCyEamaaBaaaleaacaWGZbaabeaakiaaiYcaca WHYoaacaGLOaGaayzkaaGaeyypa0JaaCOvamaaBaaaleaaceWGZbGb aebaaeqaaOGaeyOeI0IaaCOvamaaBaaaleaaceWGZbGbaebacaWGZb aabeaakiaahAfadaqhaaWcbaGaam4CaaqaaiabgkHiTiaaigdaaaGc caWHwbWaaSbaaSqaaiaadohaceWGZbGbaebaaeqaaOGaaGOlaiaabc cacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeii aiaabIcacaaIYaGaaiOlaiaaiwdacaGGPaaaaa@8410@

Remarque 1 : Il convient de souligner que, sous l'hypothèse de normalité, E( y s ¯ | y s ,β ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai aadweadaqadaqaamaaeiaabaGaaCyEamaaBaaaleaaceWGZbGbaeba aeqaaaGccaGLiWoacaWH5bWaaSbaaSqaaiaadohaaeqaaOGaaGilai aahk7aaiaawIcacaGLPaaaaaa@456C@  et V( y s ¯ | y s ,β ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai aadAfadaqadaqaamaaeiaabaGaaCyEamaaBaaaleaaceWGZbGbaeba aeqaaaGccaGLiWoacaWH5bWaaSbaaSqaaiaadohaaeqaaOGaaGilai aahk7aaiaawIcacaGLPaaaaaa@457D@  sont données, respectivement, par les membres de droite des équations de (2.5). L'EBL et sa variance associée donnés en (2.5) peuvent être considérés respectivement, comme des approximations de E( y s ¯ | y s ,β ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai aadweadaqadaqaamaaeiaabaGaaCyEamaaBaaaleaaceWGZbGbaeba aeqaaaGccaGLiWoacaWH5bWaaSbaaSqaaiaadohaaeqaaOGaaGilai aahk7aaiaawIcacaGLPaaaaaa@456C@  et V( y s ¯ | y s ,β ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai aadAfadaqadaqaamaaeiaabaGaaCyEamaaBaaaleaaceWGZbGbaeba aeqaaaGccaGLiWoacaWH5bWaaSbaaSqaaiaadohaaeqaaOGaaGilai aahk7aaiaawIcacaGLPaaaaaa@457D@  pour les cas de non-normalité.

Maintenant, si nous revenons au modèle (2.4), nous devons adapter la structure (2.1) et utiliser les résultats de (2.2) pour obtenir l'EBL de β MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai aahk7aaaa@3C55@  et sa variance associée, V ^ ( β ^ ), MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai qadAfagaqcamaabmaabaGabCOSdyaajaaacaGLOaGaayzkaaGaaiil aaaa@3F89@  donnés respectivement par :

β ^ =a+R X s ( X s R X s + V s ) 1 ( y s X s a )et V ^ ( β ^ )=C=RR X s ( X s R X s + V s ) 1 X s R.         (2.6) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai qahk7agaqcaiabg2da9iaahggacqGHRaWkcaWHsbGabCiwayaafaWa aSbaaSqaaiaadohaaeqaaOWaaeWaaeaacaWHybWaaSbaaSqaaiaado haaeqaaOGaaCOuaiqahIfagaqbamaaBaaaleaacaWGZbaabeaakiab gUcaRiaahAfadaWgaaWcbaGaam4CaaqabaaakiaawIcacaGLPaaada ahaaWcbeqaaiabgkHiTiaaigdaaaGcdaqadaqaaiaahMhadaWgaaWc baGaam4CaaqabaGccqGHsislcaWHybWaaSbaaSqaaiaadohaaeqaaO GaaCyyaaGaayjkaiaawMcaaiaaykW7caaMc8UaaeyzaiaabshacaaM c8UaaGPaVlqadAfagaqcamaabmaabaGabCOSdyaajaaacaGLOaGaay zkaaGaeyypa0JaaC4qaiabg2da9iaahkfacqGHsislcaWHsbGabCiw ayaafaWaaSbaaSqaaiaadohaaeqaaOWaaeWaaeaacaWHybWaaSbaaS qaaiaadohaaeqaaOGaaCOuaiqahIfagaqbamaaBaaaleaacaWGZbaa beaakiabgUcaRiaahAfadaWgaaWcbaGaam4CaaqabaaakiaawIcaca GLPaaadaahaaWcbeqaaiabgkHiTiaaigdaaaGccaWHybWaaSbaaSqa aiaadohaaeqaaOGaaCOuaiaai6cacaqGGaGaaeiiaiaabccacaqGGa GaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGOaGaaGOmaiaac6ca caaI2aGaaiykaaaa@80CA@

Il est facile de voir que, dans (2.6), la première équation peut être réécrite sous la forme β ^ =C( X s V s 1 y s + R 1 a ), MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai qahk7agaqcaiabg2da9iaahoeadaqadaqaaiqahIfagaqbamaaBaaa leaacaWGZbaabeaakiaahAfadaqhaaWcbaGaam4CaaqaaiabgkHiTi aaigdaaaGccaWH5bWaaSbaaSqaaiaadohaaeqaaOGaey4kaSIaaCOu amaaCaaaleqabaGaeyOeI0IaaGymaaaakiaahggaaiaawIcacaGLPa aacaGGSaaaaa@4CF7@  où C 1 = R 1 + X s V s 1 X s . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai aahoeadaahaaWcbeqaaiabgkHiTiaaigdaaaGccqGH9aqpcaWHsbWa aWbaaSqabeaacqGHsislcaaIXaaaaOGaey4kaSIabCiwayaafaWaaS baaSqaaiaadohaaeqaaOGaaCOvamaaDaaaleaacaWGZbaabaGaeyOe I0IaaGymaaaakiaahIfadaWgaaWcbaGaam4CaaqabaGccaGGUaaaaa@4AF6@  Il convient de souligner que, si nous plaçons une loi a priori vague sur β, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai aahk7acaGGSaaaaa@3D05@  en prenant R 1 0, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai aahkfadaahaaWcbeqaaiabgkHiTiaaigdaaaGccqGHsgIRcaaIWaGa aiilaaaa@4128@  nous obtenons l'estimateur par les moindres carrés minimal de β: MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai aahk7acaGG6aaaaa@3D13@   β ^ LS = ( X s V s 1 X s ) 1 X s V s 1 y s . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai qahk7agaqcamaaBaaaleaacaWGmbGaam4uaaqabaGccqGH9aqpdaqa daqaaiqahIfagaqbamaaBaaaleaacaWGZbaabeaakiaahAfadaqhaa WcbaGaam4CaaqaaiabgkHiTiaaigdaaaGccaWHybWaaSbaaSqaaiaa dohaaeqaaaGccaGLOaGaayzkaaWaaWbaaSqabeaacqGHsislcaaIXa aaaOGabCiwayaafaWaaSbaaSqaaiaadohaaeqaaOGaaCOvamaaDaaa leaacaWGZbaabaGaeyOeI0IaaGymaaaakiaahMhadaWgaaWcbaGaam 4CaaqabaGccaaIUaaaaa@534B@

Maintenant, en appliquant les propriétés bien connues des espérances et des variances conditionnelles, nous obtenons :

E[ y s ¯ | y s ]=E( E( y s ¯ | y s ,β )| y s )etV[ y s ¯ | y s ]=E( V( y s ¯ | y s ,β )| y s )+V( E( y s ¯ | y s ,β )| y s ).         (2.7) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai aadweadaWadaqaamaaeiaabaGaaCyEamaaBaaaleaaceWGZbGbaeba aeqaaaGccaGLiWoacaWH5bWaaSbaaSqaaiaadohaaeqaaaGccaGLBb GaayzxaaGaeyypa0JaamyramaabmaabaWaaqGaaeaacaWGfbWaaeWa aeaadaabcaqaaiaahMhadaWgaaWcbaGabm4CayaaraaabeaaaOGaay jcSdGaaCyEamaaBaaaleaacaWGZbaabeaakiaaiYcacaWHYoaacaGL OaGaayzkaaGaaGPaVdGaayjcSdGaaCyEamaaBaaaleaacaWGZbaabe aaaOGaayjkaiaawMcaaiaaykW7caaMc8UaaeyzaiaabshacaaMc8Ua aGPaVlaadAfadaWadaqaamaaeiaabaGaaCyEamaaBaaaleaaceWGZb GbaebaaeqaaaGccaGLiWoacaWH5bWaaSbaaSqaaiaadohaaeqaaaGc caGLBbGaayzxaaGaeyypa0JaamyramaabmaabaWaaqGaaeaacaWGwb WaaeWaaeaadaabcaqaaiaahMhadaWgaaWcbaGabm4Cayaaraaabeaa aOGaayjcSdGaaCyEamaaBaaaleaacaWGZbaabeaakiaaiYcacaWHYo aacaGLOaGaayzkaaGaaGPaVdGaayjcSdGaaCyEamaaBaaaleaacaWG ZbaabeaaaOGaayjkaiaawMcaaiabgUcaRiaadAfadaqadaqaamaaei aabaGaamyramaabmaabaWaaqGaaeaacaWH5bWaaSbaaSqaaiqadoha gaqeaaqabaaakiaawIa7aiaahMhadaWgaaWcbaGaam4CaaqabaGcca aISaGaaCOSdaGaayjkaiaawMcaaiaaykW7aiaawIa7aiaahMhadaWg aaWcbaGaam4CaaqabaaakiaawIcacaGLPaaacaGGUaGaaeiiaiaabc cacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeik aiaaikdacaGGUaGaaG4naiaacMcaaaa@97B6@

En remplaçant E( y s ¯ | y s ,β ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai aadweadaqadaqaamaaeiaabaGaaCyEamaaBaaaleaaceWGZbGbaeba aeqaaaGccaGLiWoacaWH5bWaaSbaaSqaaiaadohaaeqaaOGaaGilai aahk7aaiaawIcacaGLPaaaaaa@456C@  et V( y s ¯ | y s ,β ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai aadAfadaqadaqaamaaeiaabaGaaCyEamaaBaaaleaaceWGZbGbaeba aeqaaaGccaGLiWoacaWH5bWaaSbaaSqaaiaadohaaeqaaOGaaGilai aahk7aaiaawIcacaGLPaaaaaa@457D@  dans (2.7) par leur EBL respectif donné en (2.5) et puis, en remplaçant E( β| y s ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai aadweadaqadaqaamaaeiaabaGaaCOSdiaaykW7aiaawIa7aiaahMha daWgaaWcbaGaam4CaaqabaaakiaawIcacaGLPaaaaaa@43F9@  et V( β| y s ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai aadAfadaqadaqaamaaeiaabaGaaCOSdiaaykW7aiaawIa7aiaahMha daWgaaWcbaGaam4CaaqabaaakiaawIcacaGLPaaaaaa@440A@  par β ^ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai qahk7agaqcaaaa@3C65@  et V ^ ( β ^ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai qadAfagaqcamaabmaabaGabCOSdyaajaaacaGLOaGaayzkaaaaaa@3ED9@  donnés en (2.6), nous obtenons l'EBL de E[ y s ¯ | y s ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai aadweadaWadaqaamaaeiaabaGaaCyEamaaBaaaleaaceWGZbGbaeba aeqaaaGccaGLiWoacaWH5bWaaSbaaSqaaiaadohaaeqaaaGccaGLBb Gaayzxaaaaaa@43E1@  et sa variance associée sous la forme :

E ^ [ y s ¯ | y s ]= X s ¯ β ^ + V s ¯ s V s 1 ( y s X s β ^ )et V ^ [ y s ¯ | y s ]= V s ¯ V s ¯ s V s 1 V s s ¯ +( X s ¯ V s ¯ s V s 1 X s )C( X s ¯ V s ¯ s V s 1 X s ).         (2.8) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOabaq qabaGabmyrayaajaWaamWaaeaadaabcaqaaiaahMhadaWgaaWcbaGa bm4CayaaraaabeaaaOGaayjcSdGaaCyEamaaBaaaleaacaWGZbaabe aaaOGaay5waiaaw2faaiabg2da9iaahIfadaWgaaWcbaGabm4Cayaa raaabeaakiqahk7agaqcaiabgUcaRiaahAfadaWgaaWcbaGabm4Cay aaraGaam4CaaqabaGccaWHwbWaa0baaSqaaiaadohaaeaacqGHsisl caaIXaaaaOWaaeWaaeaacaWH5bWaaSbaaSqaaiaadohaaeqaaOGaey OeI0IaaCiwamaaBaaaleaacaWGZbaabeaakiqahk7agaqcaaGaayjk aiaawMcaaiaaykW7caaMc8UaaeyzaiaabshaaeaaceWGwbGbaKaada WadaqaamaaeiaabaGaaCyEamaaBaaaleaaceWGZbGbaebaaeqaaaGc caGLiWoacaWH5bWaaSbaaSqaaiaadohaaeqaaaGccaGLBbGaayzxaa Gaeyypa0JaaCOvamaaBaaaleaaceWGZbGbaebaaeqaaOGaeyOeI0Ia aCOvamaaBaaaleaaceWGZbGbaebacaWGZbaabeaakiaahAfadaqhaa WcbaGaam4CaaqaaiabgkHiTiaaigdaaaGccaWHwbWaaSbaaSqaaiaa dohaceWGZbGbaebaaeqaaOGaey4kaSYaaeWaaeaacaWHybWaaSbaaS qaaiqadohagaqeaaqabaGccqGHsislcaWHwbWaaSbaaSqaaiqadoha gaqeaiaadohaaeqaaOGaaCOvamaaDaaaleaacaWGZbaabaGaeyOeI0 IaaGymaaaakiaahIfadaWgaaWcbaGaam4CaaqabaaakiaawIcacaGL PaaacaWHdbWaaeWaaeaacaWHybWaaSbaaSqaaiqadohagaqeaaqaba GccqGHsislcaWHwbWaaSbaaSqaaiqadohagaqeaiaadohaaeqaaOGa aCOvamaaDaaaleaacaWGZbaabaGaeyOeI0IaaGymaaaakiaahIfada WgaaWcbaGaam4CaaqabaaakiaawIcacaGLPaaaiiaacqWFYaIOcaaI UaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabc cacaqGGaGaaeikaiaaikdacaGGUaGaaGioaiaacMcaaaaa@9C40@

Remarque 2 : De manière analogue à la remarque 1, sous l'hypothèse de normalité, nous avons que les membres de droite des équations (2.8) sont, respectivement, les valeurs de E[ y s ¯ | y s ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai aadweadaWadaqaamaaeiaabaGaaCyEamaaBaaaleaaceWGZbGbaeba aeqaaaGccaGLiWoacaWH5bWaaSbaaSqaaiaadohaaeqaaaGccaGLBb Gaayzxaaaaaa@43E1@  et V[ y s ¯ | y s ]. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai aadAfadaWadaqaamaaeiaabaGaaCyEamaaBaaaleaaceWGZbGbaeba aeqaaaGccaGLiWoacaWH5bWaaSbaaSqaaiaadohaaeqaaaGccaGLBb GaayzxaaGaaiOlaaaa@44A4@

L'expression générale de l'EBL du total T MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai aadsfaaaa@3BF0@  et de sa variance associée est obtenue en remplaçant E( y s ¯ | y s ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai aadweadaqadaqaamaaeiaabaGaaCyEamaaBaaaleaaceWGZbGbaeba aeqaaaGccaGLiWoacaWH5bWaaSbaaSqaaiaadohaaeqaaaGccaGLOa Gaayzkaaaaaa@4378@  et V( y s ¯ | y s ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai aadAfadaqadaqaamaaeiaabaGaaCyEamaaBaaaleaaceWGZbGbaeba aeqaaaGccaGLiWoacaWH5bWaaSbaaSqaaiaadohaaeqaaaGccaGLOa Gaayzkaaaaaa@4389@  dans les équations (2.3) par leurs équivalents respectifs E ^ [ y s ¯ | y s ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai qadweagaqcamaadmaabaWaaqGaaeaacaWH5bWaaSbaaSqaaiqadoha gaqeaaqabaaakiaawIa7aiaahMhadaWgaaWcbaGaam4Caaqabaaaki aawUfacaGLDbaaaaa@43F1@  et V ^ [ y s ¯ | y s ]: MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai qadAfagaqcamaadmaabaWaaqGaaeaacaWH5bWaaSbaaSqaaiqadoha gaqeaaqabaaakiaawIa7aiaahMhadaWgaaWcbaGaam4Caaqabaaaki aawUfacaGLDbaacaGG6aaaaa@44C0@

T ^ = 1 s y s + 1 s ¯ E ^ [ y s ¯ | y s ]et V ^ ( T ^ )= 1 s ¯ V ^ [ y s ¯ | y s ] 1 s ¯ .         (2.9) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai qadsfagaqcaiabg2da9iqahgdagaqbamaaBaaaleaacaWGZbaabeaa kiaahMhadaWgaaWcbaGaam4CaaqabaGccqGHRaWkceWHXaGbauaada WgaaWcbaGabm4CayaaraaabeaakiqadweagaqcamaadmaabaWaaqGa aeaacaWH5bWaaSbaaSqaaiqadohagaqeaaqabaaakiaawIa7aiaahM hadaWgaaWcbaGaam4CaaqabaaakiaawUfacaGLDbaacaaMc8UaaGPa VlaabwgacaqG0bGaaGPaVlaaykW7ceWGwbGbaKaadaqadaqaaiqads fagaqcaaGaayjkaiaawMcaaiabg2da9iqahgdagaqbamaaBaaaleaa ceWGZbGbaebaaeqaaOGabmOvayaajaWaamWaaeaadaabcaqaaiaahM hadaWgaaWcbaGabm4CayaaraaabeaaaOGaayjcSdGaaCyEamaaBaaa leaacaWGZbaabeaaaOGaay5waiaaw2faaiaahgdadaWgaaWcbaGabm 4Cayaaraaabeaakiaai6cacaqGGaGaaeiiaiaabccacaqGGaGaaeii aiaabccacaqGGaGaaeiiaiaabccacaqGOaGaaGOmaiaac6cacaaI5a Gaaiykaaaa@7053@

Il convient de souligner que, dans de nombreuses applications de (2.9), la matrice V MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai aahAfaaaa@3BF6@  est supposée être diagonale, ce qui implique que V s ¯ s =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai aahAfadaWgaaWcbaGabm4CayaaraGaam4CaaqabaGccqGH9aqpcaWH Waaaaa@3FF3@  et nous avons alors :

T ^ = 1 s y s + 1 s ¯ X s ¯ β ^ et V ^ ( T ^ )= 1 s ¯ [ V s ¯ + X s ¯ C X s ¯ ] 1 s ¯ .         (2.10) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai qadsfagaqcaiabg2da9iqahgdagaqbamaaBaaaleaacaWGZbaabeaa kiaahMhadaWgaaWcbaGaam4CaaqabaGccqGHRaWkceWHXaGbauaada WgaaWcbaGabm4CayaaraaabeaakiaahIfadaWgaaWcbaGabm4Cayaa raaabeaakiqahk7agaqcaiaaykW7caaMc8UaaeyzaiaabshacaaMc8 UaaGPaVlqadAfagaqcamaabmaabaGabmivayaajaaacaGLOaGaayzk aaGaeyypa0JabCymayaafaWaaSbaaSqaaiqadohagaqeaaqabaGcda WadaqaaiaahAfadaWgaaWcbaGabm4CayaaraaabeaakiabgUcaRiaa hIfadaWgaaWcbaGabm4CayaaraaabeaakiaahoeaceWHybGbauaada WgaaWcbaGabm4CayaaraaabeaaaOGaay5waiaaw2faaiaahgdadaWg aaWcbaGabm4Cayaaraaabeaakiaai6cacaqGGaGaaeiiaiaabccaca qGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGOaGaaGOmaiaa c6cacaaIXaGaaGimaiaacMcaaaa@6CD4@

En guise d'illustration, nous considérons certains exemples discutés par O'Hagan (1985) et proposons un nouvel estimateur par le ratio, qui est l'une des contributions des présents travaux. Tous les exemples peuvent être traités comme des cas particuliers du modèle (2.4).

2.3  Retour sur certains plans de sondage fréquents

2.3.1 Échantillonnage aléatoire simple sans remise : échangeabilité

O'Hagan (1985) a examiné le cas simple où la population ne présente aucune structure pertinente, ce qui peut se faire en spécifiant :

E( y i )=m,V( y i )=vetCov( y i , y j )=c,i,j=1,,N,ij.         (2.11) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai aadweadaqadaqaaiaadMhadaWgaaWcbaGaamyAaaqabaaakiaawIca caGLPaaacqGH9aqpcaWGTbGaaGilaiaadAfadaqadaqaaiaadMhada WgaaWcbaGaamyAaaqabaaakiaawIcacaGLPaaacqGH9aqpcaWG2bGa aGPaVlaaykW7caqGLbGaaeiDaiaaykW7caaMc8Uaae4qaiaab+gaca qG2bWaaeWaaeaacaWG5bWaaSbaaSqaaiaadMgaaeqaaOGaaGilaiaa dMhadaWgaaWcbaGaamOAaaqabaaakiaawIcacaGLPaaacqGH9aqpca WGJbGaaGilaiaadMgacaaISaGaamOAaiabg2da9iaaigdacaaISaGa eSOjGSKaaGilaiaad6eacaaISaGaeyiaIiIaamyAaiabgcMi5kaadQ gacaaIUaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeii aiaabccacaqGGaGaaeikaiaaikdacaGGUaGaaGymaiaaigdacaGGPa aaaa@7433@

Remarque 3 : On peut justifier la corrélation introduite dans le modèle (2.11) en invoquant l'imitation de l'échantillonnage aléatoire simple sans remise.

En appliquant le résultat général établi dans (2.10) à (2.11) avec β MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai aahk7aaaa@3C55@  de dimension 1, X= 1 N , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai aahIfacqGH9aqpcaWHXaWaaSbaaSqaaiaad6eaaeqaaOGaaiilaaaa @3F71@   a=m, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai aahggacqGH9aqpcaWGTbGaaiilaaaa@3EA9@   R=c MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai aahkfacqGH9aqpcaWGJbaaaa@3DE0@  et V= σ 2 I, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai aahAfacqGH9aqpcqaHdpWCdaahaaWcbeqaaiaaikdaaaGccaWHjbGa aiilaaaa@4134@  où σ 2 =vc, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai abeo8aZnaaCaaaleqabaGaaGOmaaaakiabg2da9iaadAhacqGHsisl caWGJbGaaiilaaaa@4253@  nous obtenons l'EBL de T MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai aadsfaaaa@3BF0@  et sa variance associée :

T ^ srs =n y ¯ s +( Nn ) μ ^ et V ^ ( T ^ srs )=( Nn ) σ 2 + ( Nn ) 2 c σ 2 ( σ 2 +nc ) 1 ,         (2.12) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai qadsfagaqcamaaBaaaleaacaWGZbGaamOCaiaadohaaeqaaOGaeyyp a0JaamOBaiqadMhagaqeamaaBaaaleaacaWGZbaabeaakiabgUcaRm aabmaabaGaamOtaiabgkHiTiaad6gaaiaawIcacaGLPaaacuaH8oqB gaqcaiaaykW7caaMc8UaaeyzaiaabshacaaMc8UaaGPaVlqadAfaga qcamaabmaabaGabmivayaajaWaaSbaaSqaaiaadohacaWGYbGaam4C aaqabaaakiaawIcacaGLPaaacqGH9aqpdaqadaqaaiaad6eacqGHsi slcaWGUbaacaGLOaGaayzkaaGaeq4Wdm3aaWbaaSqabeaacaaIYaaa aOGaey4kaSYaaeWaaeaacaWGobGaeyOeI0IaamOBaaGaayjkaiaawM caamaaCaaaleqabaGaaGOmaaaakiaadogacqaHdpWCdaahaaWcbeqa aiaaikdaaaGcdaqadaqaaiabeo8aZnaaCaaaleqabaGaaGOmaaaaki abgUcaRiaad6gacaWGJbaacaGLOaGaayzkaaWaaWbaaSqabeaacqGH sislcaaIXaaaaOGaaGilaiaabccacaqGGaGaaeiiaiaabccacaqGGa GaaeiiaiaabccacaqGGaGaaeiiaiaabIcacaaIYaGaaiOlaiaaigda caaIYaGaaiykaaaa@7DF3@

y ¯ s = n 1 1 s y s estlamoyennedéchantillon, μ ^ =ω y ¯ s +( 1ω )mestlavaleurprévuedesvaleursnonobservéesdeyet ω= n σ 2 c 1 +n σ 2 , σ 2 =vc. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOabaq qabaGabmyEayaaraWaaSbaaSqaaiaadohaaeqaaOGaeyypa0JaamOB amaaCaaaleqabaGaeyOeI0IaaGymaaaakiqahgdagaqbamaaBaaale aacaWGZbaabeaakiaahMhadaWgaaWcbaGaam4CaaqabaGccaaMc8Ua aGPaVlaabwgacaqGZbGaaeiDaiaaykW7caaMc8UaaeiBaiaabggaca aMc8UaaGPaVlaab2gacaqGVbGaaeyEaiaabwgacaqGUbGaaeOBaiaa bwgacaaMc8UaaGPaVlaabsgaieaacaWFzaIaaey6aiaabogacaqGOb Gaaeyyaiaab6gacaqG0bGaaeyAaiaabYgacaqGSbGaae4Baiaab6ga caaISaaabaGafqiVd0MbaKaacqGH9aqpcqaHjpWDceWG5bGbaebada WgaaWcbaGaam4CaaqabaGccqGHRaWkdaqadaqaaiaaigdacqGHsisl cqaHjpWDaiaawIcacaGLPaaacaWGTbGaaGPaVlaaykW7caqGLbGaae 4CaiaabshacaaMc8UaaGPaVlaabYgacaqGHbGaaGPaVlaaykW7caqG 2bGaaeyyaiaabYgacaqGLbGaaeyDaiaabkhacaaMc8UaaGPaVlaabc hacaqGYbGaaey6aiaabAhacaqG1bGaaeyzaiaaykW7caaMc8Uaaeiz aiaabwgacaqGZbGaaGPaVlaaykW7caqG2bGaaeyyaiaabYgacaqGLb GaaeyDaiaabkhacaqGZbGaaGPaVlaaykW7caqGUbGaae4Baiaab6ga caaMc8UaaGPaVlaab+gacaqGIbGaae4CaiaabwgacaqGYbGaaeODai aabMoacaqGLbGaae4CaiaaykW7caaMc8UaaeizaiaabwgacaaMc8Ua aGPaVlaahMhacaaMc8UaaGPaVlaabwgacaqG0baabaGaeqyYdCNaey ypa0ZaaSaaaeaacaWGUbGaeq4Wdm3aaWbaaSqabeaacqGHsislcaaI YaaaaaGcbaGaam4yamaaCaaaleqabaGaeyOeI0IaaGymaaaakiabgU caRiaad6gacqaHdpWCdaahaaWcbeqaaiabgkHiTiaaikdaaaaaaOGa aGilaiaab+gacaqG5dGaaGPaVlaaykW7cqaHdpWCdaahaaWcbeqaai aaikdaaaGccqGH9aqpcaWG2bGaeyOeI0Iaam4yaiaai6caaaaa@E122@

Il convient de souligner que μ ^ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai qbeY7aTzaajaaaaa@3CDD@  est une moyenne pondérée de la moyenne a priori m MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai aad2gaaaa@3C09@  et de la moyenne d'échantillon y ¯ s , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai qadMhagaqeamaaBaaaleaacaWGZbaabeaakiaacYcaaaa@3E0B@  où ω MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai abeM8a3baa@3CE4@  est le ratio entre les deux quantités de population. La moyenne m MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai aad2gaaaa@3C09@  peut être considérée comme le prior du chercheur pour la moyenne de population réelle y ¯ . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai qadMhagaqeaiaac6caaaa@3CDF@  L'incertitude au sujet de y i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai aadMhadaWgaaWcbaGaamyAaaqabaaaaa@3D2F@  est divisée en deux composantes : l'incertitude au sujet du niveau global des y i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai aadMhadaWgaaWcbaGaamyAaaqabaaaaa@3D2F@  (inter-variation) et l'incertitude quant à la part de chaque y i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai aadMhadaWgaaWcbaGaamyAaaqabaaaaa@3D2F@  qui peut différer de ce niveau global (intra-variation). Une mesure utile de la variabilité des unités dans la population est donnée par

S 2 = 1 N1 i=1 N ( y i y ¯ ) 2 . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai aadofadaahaaWcbeqaaiaaikdaaaGccqGH9aqpdaWcaaqaaiaaigda aeaacaWGobGaeyOeI0IaaGymaaaadaaeWbqaamaabmaabaGaamyEam aaBaaaleaacaWGPbaabeaakiabgkHiTiqadMhagaqeaaGaayjkaiaa wMcaamaaCaaaleqabaGaaGOmaaaaaeaacaWGPbGaeyypa0JaaGymaa qaaiaad6eaa0GaeyyeIuoakiaac6caaaa@4E3A@

Il n'est pas difficile de montrer que E( S 2 )=vc= σ 2 . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai aadweadaqadaqaaiaadofadaahaaWcbeqaaiaaikdaaaaakiaawIca caGLPaaacqGH9aqpcaWG2bGaeyOeI0Iaam4yaiabg2da9iabeo8aZn aaCaaaleqabaGaaGOmaaaakiaac6caaaa@4779@  Par conséquent, σ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai abeo8aZnaaCaaaleqabaGaaGOmaaaaaaa@3DC3@  peut être interprétée comme une estimation a priori de la variabilité à l'intérieur de la population. Nous obtenons aussi V( y ¯ )=c+ N 1 σ 2 . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai aadAfadaqadaqaaiqadMhagaqeaaGaayjkaiaawMcaaiabg2da9iaa dogacqGHRaWkcaWGobWaaWbaaSqabeaacqGHsislcaaIXaaaaOGaeq 4Wdm3aaWbaaSqabeaacaaIYaaaaOGaaiOlaaaa@477B@  Dans de nombreuses applications, N MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai aad6eaaaa@3BEA@  est grand et la constante  c MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai aadogaaaa@3BFF@ peut donc être considérée comme l'inter-variation.

En posant que v MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai aadAhacqGHsgIRcqGHEisPaaa@3F70@  et en maintenant σ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai abeo8aZnaaCaaaleqabaGaaGOmaaaaaaa@3DC3@  fixe, c'est-à-dire en supposant que l'on ne connaît pas les priors, les estimations dans (2.12) donnent :

T ^ srs =N y ¯ s et V ^ ( T ^ srs )= N 2 ( 1 n N ) σ 2 n . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaau aabeqabeaaaeaaceWGubGbaKaadaWgaaWcbaGaam4CaiaadkhacaWG Zbaabeaakiabg2da9iaad6eacaaMc8UabmyEayaaraWaaSbaaSqaai aadohaaeqaaOGaaGjcVlaaysW7caqGLbGaaeiDaiaaysW7caaMi8Ua bmOvayaajaWaaeWaaeaaceWGubGbaKaadaWgaaWcbaGaam4Caiaadk hacaWGZbaabeaaaOGaayjkaiaawMcaaiabg2da9iaad6eadaahaaWc beqaaiaaikdaaaGcdaqadaqaaiaaigdacqGHsisldaWcaaqaaiaad6 gaaeaacaWGobaaaaGaayjkaiaawMcaamaalaaabaGaeq4Wdm3aaWba aSqabeaacaaIYaaaaaGcbaGaamOBaaaacaaIUaaaaaaa@5FA9@

Ces expressions sont fort semblables à l'estimation bien connue du total et à sa variance dans le contexte fondé sur le plan de sondage pour le cas de l'échantillonnage aléatoire simple. O'Hagan (1985) a discuté de certains moyens possibles d'éviter la tâche difficile d'attribuer une valeur à σ 2 . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai abeo8aZnaaCaaaleqabaGaaGOmaaaakiaac6caaaa@3E7F@  Le moyen le plus naturel de le faire consiste à trouver son EBL, mais linéaire en les carrés et les termes de variance des produits croisés. Cependant, il est nécessaire de spécifier les moments d'ordre quatre des y i . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai aadMhadaWgaaWcbaGaamyAaaqabaGccaqGUaaaaa@3DEA@  Goldstein (1979) a proposé un EBL de la variance qui n'utilise que des fonctions linéaires des données. Néanmoins, on obtient une expression compliquée de la variance associée de son EBL modifié. O'Hagan (1985) a soutenu que, si l'information a priori au sujet des composantes de la variance est faible, toute estimation a posteriori s'approche des estimations non bayésiennes classiques obtenues en utilisant uniquement les données, lorsque ce gendre d'estimations est disponible. Par conséquent, il a proposé, en guise de procédure bayésienne approximative, d'introduire ces estimations de variance classiques par substitution dans l'EBL et dans sa variance associée lorsque cela est approprié. Dans le cas qui nous occupe, nous pouvons remplacer σ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai abeo8aZnaaCaaaleqabaGaaGOmaaaaaaa@3DC3@  par s 2 = ( n1 ) 1 i=1 n ( y i y ¯ s ) 2 , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai aadohadaahaaWcbeqaaiaaikdaaaGccqGH9aqpdaqadaqaaiaad6ga cqGHsislcaaIXaaacaGLOaGaayzkaaWaaWbaaSqabeaacqGHsislca aIXaaaaOWaaabmaeqaleaacaWGPbGaeyypa0JaaGymaaqaaiaad6ga a0GaeyyeIuoakmaabmaabaGaamyEamaaBaaaleaacaWGPbaabeaaki abgkHiTiqadMhagaqeamaaBaaaleaacaWGZbaabeaaaOGaayjkaiaa wMcaamaaCaaaleqabaGaaGOmaaaakiaacYcaaaa@5239@  qui est sans biais sous le plan pour S 2 . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai aadofadaahaaWcbeqaaiaaikdaaaGccaGGUaaaaa@3D94@

2.3.2 Échantillonnage aléatoire simple stratifié sans remise

Soit y hi MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai aadMhadaWgaaWcbaGaamiAaiaadMgaaeqaaaaa@3E1C@  la i e MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai aadMgadaahaaWcbeqaaiaabwgaaaaaaa@3D1A@  unité, i=1,..., N h MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai aadMgacqGH9aqpcaaIXaGaaGilaiaai6cacaaIUaGaaGOlaiaaiYca caWGobWaaSbaaSqaaiaadIgaaeqaaaaa@4346@  appartenant à la strate h, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai aadIgacaGGSaaaaa@3CB4@   h=1,..,H. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai aadIgacqGH9aqpcaaIXaGaaGilaiaai6cacaaIUaGaaGilaiaadIea caGGUaaaaa@4220@  Nous supposons que la taille de strate, N h , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai aad6eadaWgaaWcbaGaamiAaaqabaGccaGGSaaaaa@3DBD@  est connue pour toutes les strates. L'échangeabilité d'ordre deux dans chaque strate est énoncée dans O'Hagan (1985) sous la forme :

E( y hi )= m h ,V( y hi )= v h ,Cov( y hi , y hj )= c h ,ijetCov( y hi , y lj )= d hl ,hl. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai aadweadaqadeqaaiaadMhadaWgaaWcbaGaamiAaiaadMgaaeqaaaGc caGLOaGaayzkaaGaeyypa0JaamyBamaaBaaaleaacaWGObaabeaaki aaiYcacaWGwbWaaeWaaeaacaWG5bWaaSbaaSqaaiaadIgacaWGPbaa beaaaOGaayjkaiaawMcaaiabg2da9iaadAhadaWgaaWcbaGaamiAaa qabaGccaaISaGaae4qaiaab+gacaqG2bWaaeWabeaacaWG5bWaaSba aSqaaiaadIgacaWGPbaabeaakiaaiYcacaWG5bWaaSbaaSqaaiaadI gacaWGQbaabeaaaOGaayjkaiaawMcaaiabg2da9iaadogadaWgaaWc baGaamiAaaqabaGccaaISaGaamyAaiabgcMi5kaadQgacaaMc8UaaG PaVlaabwgacaqG0bGaaGPaVlaaykW7caqGdbGaae4BaiaabAhadaqa deqaaiaadMhadaWgaaWcbaGaamiAaiaadMgaaeqaaOGaaGilaiaadM hadaWgaaWcbaGaamiBaiaadQgaaeqaaaGccaGLOaGaayzkaaGaeyyp a0JaamizamaaBaaaleaacaWGObGaamiBaaqabaGccaaISaGaamiAai abgcMi5kaadYgacaaIUaaaaa@7C48@

Remarque 4 : Il est raisonnable de supposer que l'information obtenue au sujet d'une strate pourrait modifier les croyances au sujet des autres strates dans certaines applications spéciales. Cependant, si nous voulons imiter l'échantillonnage aléatoire simple stratifié, nous devons supposer que les observations dans les diverses strates ne sont pas corrélées, en posant que d hl =0. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai aadsgadaWgaaWcbaGaamiAaiaadYgaaeqaaOGaeyypa0JaaGimaiaa c6caaaa@4086@

Le modèle général (2.4) peut être appliqué à ce cas en prenant X=diag( X 1 ,, X H ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai aahIfacqGH9aqpcaqGKbGaaeyAaiaabggacaqGNbWaaeWaaeaacaWH ybWaaSbaaSqaaiaaigdaaeqaaOGaaGilaiablAciljaaiYcacaWHyb WaaSbaaSqaaiaadIeaaeqaaaGccaGLOaGaayzkaaaaaa@486C@  et V=diag( V 1 ,, V H ), MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai aahAfacqGH9aqpcaqGKbGaaeyAaiaabggacaqGNbWaaeWaaeaacaWH wbWaaSbaaSqaaiaaigdaaeqaaOGaaGilaiablAciljaaiYcacaWHwb WaaSbaaSqaaiaadIeaaeqaaaGccaGLOaGaayzkaaGaaiilaaaa@4916@  avec X h = 1 N h MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai aahIfadaWgaaWcbaGaamiAaaqabaGccqGH9aqpcaWHXaWaaSbaaSqa aiaad6eadaWgaaqaaiaadIgaaeqaaaqabaaaaa@40E8@  et V h = σ h 2 I N h , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai aahAfadaWgaaWcbaGaamiAaaqabaGccqGH9aqpcqaHdpWCdaqhaaWc baGaamiAaaqaaiaaikdaaaGccaWHjbWaaSbaaSqaaiaad6eadaWgaa qaaiaadIgaaeqaaaqabaGccaGGSaaaaa@455B@  où σ h 2 = v h c h , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai abeo8aZnaaDaaaleaacaWGObaabaGaaGOmaaaakiabg2da9iaadAha daWgaaWcbaGaamiAaaqabaGccqGHsislcaWGJbWaaSbaaSqaaiaadI gaaeqaaOGaaiilaaaa@4586@   MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai abgcGiIaaa@3BE7@   h=1,,H, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai aadIgacqGH9aqpcaaIXaGaaGilaiablAciljaaiYcacaWGibGaaGil aaaa@41D6@   a=( m 1 ,, m H ), MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai aahggacqGH9aqpdaqadaqaaiaad2gadaWgaaWcbaGaaGymaaqabaGc caaISaGaeSOjGSKaaGilaiaad2gadaWgaaWcbaGaamisaaqabaaaki aawIcacaGLPaaaiiaacqWFYaIOcqWFSaalaaa@4756@   R MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai aahkfaaaa@3BF2@  est une matrice de dimensions H×H MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai aadIeacqGHxdaTcaWGibaaaa@3EC8@  avec R hl = c h , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai aadkfadaWgaaWcbaGaamiAaiaadYgaaeqaaOGaeyypa0Jaam4yamaa BaaaleaacaWGObaabeaakiaacYcaaaa@41C3@  si h=l MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai aadIgacqGH9aqpcaWGSbaaaa@3DFB@  et R hl = d hl MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai aadkfadaWgaaWcbaGaamiAaiaadYgaaeqaaOGaeyypa0Jaamizamaa BaaaleaacaWGObGaamiBaaqabaaaaa@41FB@  autrement. L'EBL de T MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbcvPDwzYbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x e9LqFf0xe9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qq aq=Jf9sr0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaai aadsfaaaa@3BF0@  et sa variance associée sont obtenus au moyen de (2.10) et figurent dans O'Hagan (1985). Les modèles pour l'échantillonnage en grappes sont donnés dans Bolfarine et Zacks (1992), page 11. L'EBL des modèles avec grappes figurent dans O'Hagan (1985).

Précédent | Suivant

Signaler un problème sur cette page

Quelque chose ne fonctionne pas? L'information n'est plus à jour? Vous ne trouvez pas ce que vous cherchez?

S'il vous plaît contactez-nous et nous informer comment nous pouvons vous aider.

Avis de confidentialité

Date de modification :