4 Log-vraisemblance composite pondérée : une approche unifiée

J.N.K. Rao, F. Verret et M.A. Hidiroglou

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À la présente section, nous proposons une approche unifiée applicable aux modèles multiniveaux linéaires ainsi que linéaires généralisés. Cette approche est fondée sur le concept de la vraisemblance composite qui a acquis de la popularité dans la littérature ne portant pas sur les sondages pour traiter les données en grappes ou les données spatiales (voir p. ex., Lindsay 1988, Lele et Taper 2002 et Varin, Reid et Firth 2011). Une vraisemblance composite marginale par paire s'obtient en multipliant les contributions à la vraisemblance de toutes les paires distinctes dans les grappes. Notons que la vraisemblance composite est obtenue en prétendant que les sous-modèles sont indépendants. Lorsque le modèle de superpopulation est vérifié pour l'échantillon, nous pouvons obtenir les estimateurs des paramètres en maximisant la vraisemblance composite par paire. Ici, nous étendons cette approche aux plans de sondage informatifs en obtenant des équations d'estimation pondérées qui requièrent seulement les poids marginaux w i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFysPYR0xH8viFfea0dXdHaVhbvd9G8Grpq0xc9vqpq 0xbba9q8WqFfeaY=biNkVc=He9pgeu0dXdar=Jir=JbbG8Fq0xfr=x frVcFf0dmeaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWG3bWaaS baaSqaaiaadMgaaeqaaaaa@36F7@ et w j | i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFysPYR0xH8viFfea0dXdHaVhbvd9G8Grpq0xc9vqpq 0xbba9q8WqFfeaY=biNkVc=He9pgeu0dXdar=Jir=JbbG8Fq0xfr=x frVcFf0dmeaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWG3bWaaS baaSqaaiaadQgacaGG8bGaamyAaaqabaaaaa@38E6@ et les poids par paire w j k | i , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFysPYR0xH8viFfea0dXdHaVhbvd9G8Grpq0xc9vqpq 0xbba9q8WqFfeaY=biNkVc=He9pgeu0dXdar=Jir=JbbG8Fq0xfr=x frVcFf0dmeaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWG3bWaaS baaSqaaiaadQgacaWGRbGaaiiFaiaadMgaaeqaaOGaaiilaaaa@3A90@ comme à la section 3.

La log vraisemblance composite par paire de recensement est donnée par

l C ( θ ) = i = 1 N j < k = 1 M i log f ( y i j , y i k | θ ) ,        ( 4.1 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFysPYR0xH8viFfea0dXdHaVhbvd9G8Grpq0xc9vqpq 0xbba9q8WqFfeaY=biNkVc=He9pgeu0dXdar=Jir=JbbG8Fq0xfr=x frVcFf0dmeaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGSbWaaS baaSqaaiaadoeaaeqaaOGaaiikaiaahI7acaGGPaGaeyypa0ZaaabC aeaadaaeWbqaaiGacYgacaGGVbGaai4zaiaadAgadaqadaqaamaaei aabaGaamyEamaaBaaaleaacaWGPbGaamOAaaqabaGccaGGSaGaaGjb VlaaykW7caWG5bWaaSbaaSqaaiaadMgacaWGRbaabeaaaOGaayjcSd GaaGPaVlaahI7aaiaawIcacaGLPaaacaGGSaaaleaacaWGQbGaeyip aWJaam4Aaiabg2da9iaaigdaaeaacaWGnbWaaSbaaWqaaiaadMgaae qaaaqdcqGHris5aaWcbaGaamyAaiabg2da9iaaigdaaeaacaWGobaa niabggHiLdGccaWLjaGaaCzcamaabmaabaaeaaaaaaaaa8qacaaI0a GaaiOlaiaaigdaa8aacaGLOaGaayzkaaaaaa@6280@

f ( y i j , y i k | θ ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFysPYR0xH8viFfea0dXdHaVhbvd9G8Grpq0xc9vqpq 0xbba9q8WqFfeaY=biNkVc=He9pgeu0dXdar=Jir=JbbG8Fq0xfr=x frVcFf0dmeaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGMbWaae WaaeaadaabcaqaaiaadMhadaWgaaWcbaGaamyAaiaadQgaaeqaaOGa aiilaiaaysW7caaMc8UaamyEamaaBaaaleaacaWGPbGaam4Aaaqaba aakiaawIa7aiaahI7aaiaawIcacaGLPaaaaaa@441B@ est la densité de probabilité conjointe marginale de y i j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFysPYR0xH8viFfea0dXdHaVhbvd9G8Grpq0xc9vqpq 0xbba9q8WqFfeaY=biNkVc=He9pgeu0dXdar=Jir=JbbG8Fq0xfr=x frVcFf0dmeaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWG5bWaaS baaSqaaiaadMgacaWGQbaabeaaaaa@37E9@ et y i k . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFysPYR0xH8viFfea0dXdHaVhbvd9G8Grpq0xc9vqpq 0xbba9q8WqFfeaY=biNkVc=He9pgeu0dXdar=Jir=JbbG8Fq0xfr=x frVcFf0dmeaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWG5bWaaS baaSqaaiaadMgacaWGRbaabeaakiaac6caaaa@38A6@ Nous estimons (4.1) par la log-vraisemblance composite par paire pondérée par les poids de sondage

l w C ( θ ) = i s w i j < k s ( i ) w j k | i log f ( y i j , y i k | θ )        ( 4.2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFysPYR0xH8viFfea0dXdHaVhbvd9G8Grpq0xc9vqpq 0xbba9q8WqFfeaY=biNkVc=He9pgeu0dXdar=Jir=JbbG8Fq0xfr=x frVcFf0dmeaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGSbWaaS baaSqaaiaadEhacaWGdbaabeaakiaacIcacaWH4oGaaiykaiabg2da 9maaqafabaGaam4DamaaBaaaleaacaWGPbaabeaakmaaqafabaGaam 4DamaaBaaaleaadaabcaqaaiaadQgacaWGRbaacaGLiWoacaWGPbaa beaakiGacYgacaGGVbGaai4zaiaadAgadaqadaqaamaaeiaabaGaam yEamaaBaaaleaacaWGPbGaamOAaaqabaGccaGGSaGaaGPaVlaaykW7 caWG5bWaaSbaaSqaaiaadMgacaWGRbaabeaaaOGaayjcSdGaaCiUda GaayjkaiaawMcaaaWcbaGaamOAaiabgYda8iaadUgacqGHiiIZcaWG ZbGaaiikaiaadMgacaGGPaaabeqdcqGHris5aaWcbaGaamyAaiabgI GiolaadohaaeqaniabggHiLdGccaWLjaGaaCzcamaabmaabaaeaaaa aaaaa8qacaaI0aGaaiOlaiaaikdaa8aacaGLOaGaayzkaaaaaa@69B3@

qui dépend seulement des probabilités d'inclusion de niveau 1 et de niveau 2 d'ordre 1 et de probabilités d'inclusion de niveau 1 d'ordre 2. Puis, nous résolvons les équations de score composite pondérées

U ^ w C ( θ ) = l w C ( θ ) / θ = 0 ,        ( 4.3 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFysPYR0xH8viFfea0dXdHaVhbvd9G8Grpq0xc9vqpq 0xbba9q8WqFfeaY=biNkVc=He9pgeu0dXdar=Jir=JbbG8Fq0xfr=x frVcFf0dmeaabaqaciaacaGaaeqabaWaaeaaeaaakeaaceWHvbGbaK aadaWgaaWcbaGaam4DaiaadoeaaeqaaOGaaiikaiaahI7acaGGPaGa eyypa0ZaaSGbaeaacqGHciITcaWGSbWaaSbaaSqaaiaadEhacaWGdb aabeaakiaacIcacaWH4oGaaiykaaqaaiabgkGi2kaahI7aaaGaeyyp a0JaaCimaiaacYcacaWLjaGaaCzcamaabmaabaaeaaaaaaaaa8qaca aI0aGaaiOlaiaaiodaa8aacaGLOaGaayzkaaaaaa@4CB2@

provenant de (4.2) pour obtenir un estimateur de la vraisemblance composite pondérée, θ ^ w C , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFysPYR0xH8viFfea0dXdHaVhbvd9G8Grpq0xc9vqpq 0xbba9q8WqFfeaY=biNkVc=He9pgeu0dXdar=Jir=JbbG8Fq0xfr=x frVcFf0dmeaabaqaciaacaGaaeqabaWaaeaaeaaakeaaceWH4oGbaK aadaWgaaWcbaGaam4DaiaadoeaaeqaaOGaaiilaaaa@38DF@ de θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFysPYR0xH8viFfea0dXdHaVhbvd9G8Grpq0xc9vqpq 0xbba9q8WqFfeaY=biNkVc=He9pgeu0dXdar=Jir=JbbG8Fq0xfr=x frVcFf0dmeaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWH4oaaaa@3625@ . La méthode proposée est applicable aux modèles à deux niveaux linéaires et linéaires généralisés.

Nous notons que U ^ w C ( θ ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFysPYR0xH8viFfea0dXdHaVhbvd9G8Grpq0xc9vqpq 0xbba9q8WqFfeaY=biNkVc=He9pgeu0dXdar=Jir=JbbG8Fq0xfr=x frVcFf0dmeaabaqaciaacaGaaeqabaWaaeaaeaaakeaaceWHvbGbaK aadaWgaaWcbaGaam4DaiaadoeaaeqaaOGaaiikaiaahI7acaGGPaaa aa@3A67@ , donné par (4.3), est un vecteur de fonctions d'estimation d'espérance nulle par rapport au plan et au modèle, c.-à-d. E m E p { U ^ w C ( θ ) } = 0 . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFysPYR0xH8viFfea0dXdHaVhbvd9G8Grpq0xc9vqpq 0xbba9q8WqFfeaY=biNkVc=He9pgeu0dXdar=Jir=JbbG8Fq0xfr=x frVcFf0dmeaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGfbWaaS baaSqaaiaad2gaaeqaaOGaamyramaaBaaaleaacaWGWbaabeaakiaa ykW7daGadaqaaiqahwfagaqcamaaBaaaleaacaWG3bGaam4qaaqaba GccaGGOaGaaCiUdiaacMcaaiaawUhacaGL9baacaaMc8UaaGjbVlab g2da9iaahcdacaGGUaaaaa@4792@ En utilisant ce résultat, on peut montrer que l'estimateur de la vraisemblance composite pondérée (VCP) θ ^ w C MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFysPYR0xH8viFfea0dXdHaVhbvd9G8Grpq0xc9vqpq 0xbba9q8WqFfeaY=biNkVc=He9pgeu0dXdar=Jir=JbbG8Fq0xfr=x frVcFf0dmeaabaqaciaacaGaaeqabaWaaeaaeaaakeaaceWH4oGbaK aadaWgaaWcbaGaam4Daiaadoeaaeqaaaaa@3826@ de θ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFysPYR0xH8viFfea0dXdHaVhbvd9G8Grpq0xc9vqpq 0xbba9q8WqFfeaY=biNkVc=He9pgeu0dXdar=Jir=JbbG8Fq0xfr=x frVcFf0dmeaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWH4oaaaa@3626@ est convergent sous le modèle quand le nombre d'unités de niveau 2 dans l'échantillon, n , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFysPYR0xH8viFfea0dXdHaVhbvd9G8Grpq0xc9vqpq 0xbba9q8WqFfeaY=biNkVc=He9pgeu0dXdar=Jir=JbbG8Fq0xfr=x frVcFf0dmeaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGUbGaai ilaaaa@3685@ augmente, même si les tailles d'échantillon dans les grappes, m i , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFysPYR0xH8viFfea0dXdHaVhbvd9G8Grpq0xc9vqpq 0xbba9q8WqFfeaY=biNkVc=He9pgeu0dXdar=Jir=JbbG8Fq0xfr=x frVcFf0dmeaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGTbWaaS baaSqaaiaadMgaaeqaaOGaaiilaaaa@37A8@ sont petites. La preuve est exposée en détail dans Yi, Rao et Li (2012). Dans le contexte ne faisant pas appel au sondage, les preuves théoriques et empiriques que l'approche de la vraisemblance composite conduit à des estimateurs efficaces sont limitées (p. ex., Bellio et Varin 2005, Lindsay et coll. 2011). Notre étude en simulation (section 5) indique que l'approche de la vraisemblance composite pondérée donne de bons résultats en ce qui concerne l'efficacité, même si les tailles d'échantillon dans les grappes sont petites.

Dans le cas du modèle à erreurs emboîtées (3.13), en nous inspirant de Lele et Taper (2002), nous pouvons simplifier l'approche de la vraisemblance composite par paire en remplaçant la densité de probabilité bivariée f ( y i j , y i k | θ ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFysPYR0xH8viFfea0dXdHaVhbvd9G8Grpq0xc9vqpq 0xbba9q8WqFfeaY=biNkVc=He9pgeu0dXdar=Jir=JbbG8Fq0xfr=x frVcFf0dmeaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGMbWaae WaaeaadaabcaqaaiaadMhadaWgaaWcbaGaamyAaiaadQgaaeqaaOGa aiilaiaadMhadaWgaaWcbaGaamyAaiaadUgaaeqaaaGccaGLiWoaca WH4oaacaGLOaGaayzkaaaaaa@4103@ par les densités de probabilité univariées de y i j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFysPYR0xH8viFfea0dXdHaVhbvd9G8Grpq0xc9vqpq 0xbba9q8WqFfeaY=biNkVc=He9pgeu0dXdar=Jir=JbbG8Fq0xfr=x frVcFf0dmeaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWG5bWaaS baaSqaaiaadMgacaWGQbaabeaaaaa@37E9@ et la différence z i j k = y i j y i k . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFysPYR0xH8viFfea0dXdHaVhbvd9G8Grpq0xc9vqpq 0xbba9q8WqFfeaY=biNkVc=He9pgeu0dXdar=Jir=JbbG8Fq0xfr=x frVcFf0dmeaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWG6bWaaS baaSqaaiaadMgacaWGQbGaam4AaaqabaGccqGH9aqpcaWG5bWaaSba aSqaaiaadMgacaWGQbaabeaakiabgkHiTiaadMhadaWgaaWcbaGaam yAaiaadUgaaeqaaOGaaiOlaaaa@41AC@ Pour le modèle de la moyenne (2.2), nous avons y i j ~ N ( μ , σ v 2 + σ e 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFysPYR0xH8viFfea0dXdHaVhbvd9G8Grpq0xc9vqpq 0xbba9q8WqFfeaY=biNkVc=He9pgeu0dXdar=Jir=JbbG8Fq0xfr=x frVcFf0dmeaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWG5bWaaS baaSqaaiaadMgacaWGQbaabeaakiaac6hacaWGobWaaeWaaeaacqaH 8oqBcaGGSaGaaGjbVlaaykW7cqaHdpWCdaqhaaWcbaGaamODaaqaai aaikdaaaGccqGHRaWkcqaHdpWCdaqhaaWcbaGaamyzaaqaaiaaikda aaaakiaawIcacaGLPaaaaaa@4902@ et z i j k ~ N ( 0 , 2 σ e 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFysPYR0xH8viFfea0dXdHaVhbvd9G8Grpq0xc9vqpq 0xbba9q8WqFfeaY=biNkVc=He9pgeu0dXdar=Jir=JbbG8Fq0xfr=x frVcFf0dmeaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWG6bWaaS baaSqaaiaadMgacaWGQbGaam4AaaqabaGccaGG+bGaamOtamaabmaa baGaaGimaiaacYcacaaMe8UaaGPaVlaaikdacqaHdpWCdaqhaaWcba GaamyzaaqaaiaaikdaaaaakiaawIcacaGLPaaaaaa@4520@ . En reparamétrisant θ = ( μ , σ v 2 , σ e 2 ) T MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFysPYR0xH8viFfea0dXdHaVhbvd9G8Grpq0xc9vqpq 0xbba9q8WqFfeaY=biNkVc=He9pgeu0dXdar=Jir=JbbG8Fq0xfr=x frVcFf0dmeaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWH4oGaey ypa0ZaaeWaaeaacqaH8oqBcaGGSaGaeq4Wdm3aa0baaSqaaiaadAha aeaacaaIYaaaaOGaaiilaiabeo8aZnaaDaaaleaacaWGLbaabaGaaG OmaaaaaOGaayjkaiaawMcaamaaCaaaleqabaGaamivaaaaaaa@4422@ de manière que ϕ = ( μ , σ 2 , σ e 2 ) T MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8HjLkVs FfYxH8vqaqpepec8Eeun0dYdg9arFj0xb9arFfea0dXdd9vqai=hGC Q8k8hs0=yqqrpepae9pIe9pgeaY=brFve9Fve9k8vqpWqaaeaabiGa aiaacaqabeaaceqaamaaaOqaaiabjw9aMjabg2da9maabmaabaGaeq iVd0Maaiilaiabeo8aZnaaCaaaleqabaGaaGOmaaaakiaacYcacqaH dpWCdaqhaaWcbaGaamyzaaqaaiaaikdaaaaakiaawIcacaGLPaaada ahaaWcbeqaaiaadsfaaaaaaa@3F75@ , où σ 2 = σ v 2 + σ e 2 , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFysPYR0xH8viFfea0dXdHaVhbvd9G8Grpq0xc9vqpq 0xbba9q8WqFfeaY=biNkVc=He9pgeu0dXdar=Jir=JbbG8Fq0xfr=x frVcFf0dmeaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHdpWCda ahaaWcbeqaaiaaikdaaaGccqGH9aqpcqaHdpWCdaqhaaWcbaGaamOD aaqaaiaaikdaaaGccqGHRaWkcqaHdpWCdaqhaaWcbaGaamyzaaqaai aaikdaaaGccaGGSaaaaa@4181@ nous voyons que les paramètres des deux densités de probabilité univariées sont distincts et que les log-vraisemblances composites correspondant à y i j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFysPYR0xH8viFfea0dXdHaVhbvd9G8Grpq0xc9vqpq 0xbba9q8WqFfeaY=biNkVc=He9pgeu0dXdar=Jir=JbbG8Fq0xfr=x frVcFf0dmeaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWG5bWaaS baaSqaaiaadMgacaWGQbaabeaaaaa@37E9@ et z i j k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFysPYR0xH8viFfea0dXdHaVhbvd9G8Grpq0xc9vqpq 0xbba9q8WqFfeaY=biNkVc=He9pgeu0dXdar=Jir=JbbG8Fq0xfr=x frVcFf0dmeaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWG6bWaaS baaSqaaiaadMgacaWGQbGaam4Aaaqabaaaaa@38DA@ sont données par

l w C y ( μ , σ 2 ) = i s w i j s ( i ) w j | i log f ( y i j | μ , σ 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFysPYR0xH8viFfea0dXdHaVhbvd9G8Grpq0xc9vqpq 0xbba9q8WqFfeaY=biNkVc=He9pgeu0dXdar=Jir=JbbG8Fq0xfr=x frVcFf0dmeaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGSbWaaS baaSqaaiaadEhacaWGdbGaamyEaaqabaGcdaqadaqaaiabeY7aTjaa cYcacqaHdpWCdaahaaWcbeqaaiaaikdaaaaakiaawIcacaGLPaaacq GH9aqpdaaeqbqaaiaadEhadaWgaaWcbaGaamyAaaqabaGcdaaeqbqa aiaadEhadaWgaaWcbaWaaqGaaeaacaWGQbaacaGLiWoacaaMc8Uaam yAaaqabaGcciGGSbGaai4BaiaacEgacaWGMbWaaeWaaeaadaabcaqa aiaadMhadaWgaaWcbaGaamyAaiaadQgaaeqaaaGccaGLiWoacqaH8o qBcaGGSaGaeq4Wdm3aaWbaaSqabeaacaaIYaaaaaGccaGLOaGaayzk aaaaleaacaWGQbGaeyicI4Saam4CaiaacIcacaWGPbGaaiykaaqab0 GaeyyeIuoaaSqaaiaadMgacqGHiiIZcaWGZbaabeqdcqGHris5aaaa @652F@

et

l w C z ( σ e 2 ) = i s w i j < k s ( i ) w j k | i log f ( z i j k | σ e 2 ) . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFysPYR0xH8viFfea0dXdHaVhbvd9G8Grpq0xc9vqpq 0xbba9q8WqFfeaY=biNkVc=He9pgeu0dXdar=Jir=JbbG8Fq0xfr=x frVcFf0dmeaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGSbWaaS baaSqaaiaadEhacaWGdbGaamOEaaqabaGcdaqadaqaaiabeo8aZnaa DaaaleaacaWGLbaabaGaaGOmaaaaaOGaayjkaiaawMcaaiabg2da9m aaqafabaGaam4DamaaBaaaleaacaWGPbaabeaakmaaqafabaGaam4D amaaBaaaleaadaabcaqaaiaadQgacaWGRbaacaGLiWoacaaMc8Uaam yAaaqabaGcciGGSbGaai4BaiaacEgacaWGMbWaaeWaaeaadaabcaqa aiaadQhadaWgaaWcbaGaamyAaiaadQgacaWGRbaabeaaaOGaayjcSd GaaGPaVlabeo8aZnaaDaaaleaacaWGLbaabaGaaGOmaaaaaOGaayjk aiaawMcaaiaac6caaSqaaiaadQgacqGH8aapcaWGRbGaeyicI4Saam 4CaiaacIcacaWGPbGaaiykaaqab0GaeyyeIuoaaSqaaiaadMgacqGH iiIZcaWGZbaabeqdcqGHris5aaaa@6849@

Nous résolvons alors le système d'équations de score composite pondérées résultantes

U ^ w C y 1 ( μ , σ 2 ) = l w C y ( μ , σ 2 ) / μ = i s w i j s i w j | i ( y i j μ ) / σ 2 = 0 , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFysPYR0xH8viFfea0dXdHaVhbvd9G8Grpq0xc9vqpq 0xbba9q8WqFfeaY=biNkVc=He9pgeu0dXdar=Jir=JbbG8Fq0xfr=x frVcFf0dmeaabaqaciaacaGaaeqabaWaaeaaeaaakeaaceWGvbGbaK aadaWgaaWcbaGaam4DaiaadoeacaWG5bGaaGymaaqabaGcdaqadaqa aiabeY7aTjaacYcacqaHdpWCdaahaaWcbeqaaiaaikdaaaaakiaawI cacaGLPaaacqGH9aqpdaWcgaqaaiabgkGi2kaadYgadaWgaaWcbaGa am4DaiaadoeacaWG5baabeaakmaabmaabaGaeqiVd0Maaiilaiabeo 8aZnaaCaaaleqabaGaaGOmaaaaaOGaayjkaiaawMcaaaqaaiabgkGi 2kabeY7aTbaacqGH9aqpdaaeqbqaaiaadEhadaWgaaWcbaGaamyAaa qabaGcdaaeqbqaaiaadEhadaWgaaWcbaWaaqGaaeaacaWGQbaacaGL iWoacaaMc8UaamyAaaqabaGcdaWcgaqaamaabmaabaGaamyEamaaBa aaleaacaWGPbGaamOAaaqabaGccqGHsislcqaH8oqBaiaawIcacaGL PaaaaeaacqaHdpWCdaahaaWcbeqaaiaaikdaaaaaaOGaeyypa0JaaG imaiaacYcaaSqaaiaadQgacqGHiiIZcaWGZbWaaSbaaWqaaiaadMga aeqaaaWcbeqdcqGHris5aaWcbaGaamyAaiabgIGiolaadohaaeqani abggHiLdaaaa@725F@

U ^ w C y 2 ( μ , σ 2 ) = l w C y ( μ , σ 2 ) / σ 2 = 1 2 i s w i j s ( i ) w j | i [ 1 σ 2 + ( y i j μ ) 2 σ 4 ] = 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFysPYR0xH8viFfea0dXdHaVhbvd9G8Grpq0xc9vqpq 0xbba9q8WqFfeaY=biNkVc=He9pgeu0dXdar=Jir=JbbG8Fq0xfr=x frVcFf0dmeaabaqaciaacaGaaeqabaWaaeaaeaaakeaaceWGvbGbaK aadaWgaaWcbaGaam4DaiaadoeacaWG5bGaaGOmaaqabaGcdaqadaqa aiabeY7aTjaacYcacqaHdpWCdaahaaWcbeqaaiaaikdaaaaakiaawI cacaGLPaaacqGH9aqpdaWcgaqaaiabgkGi2kaadYgadaWgaaWcbaGa am4DaiaadoeacaWG5baabeaakmaabmaabaGaeqiVd0Maaiilaiabeo 8aZnaaCaaaleqabaGaaGOmaaaaaOGaayjkaiaawMcaaaqaaiabgkGi 2kabeo8aZnaaCaaaleqabaGaaGOmaaaaaaGccqGH9aqpdaWcaaqaai aaigdaaeaacaaIYaaaamaaqafabaGaam4DamaaBaaaleaacaWGPbaa beaakmaaqafabaGaam4DamaaBaaaleaadaabcaqaaiaadQgaaiaawI a7aiaadMgaaeqaaOWaamWaaeaacqGHsisldaWcaaqaaiaaigdaaeaa cqaHdpWCdaqhaaWcbaaabaGaaGOmaaaaaaGccqGHRaWkdaWcaaqaam aabmaabaGaamyEamaaBaaaleaacaWGPbGaamOAaaqabaGccqGHsisl cqaH8oqBaiaawIcacaGLPaaadaahaaWcbeqaaiaaikdaaaaakeaacq aHdpWCdaahaaWcbeqaaiaaisdaaaaaaaGccaGLBbGaayzxaaGaeyyp a0JaaGimaaWcbaGaamOAaiabgIGiolaadohacaGGOaGaamyAaiaacM caaeqaniabggHiLdaaleaacaWGPbGaeyicI4Saam4Caaqab0Gaeyye Iuoaaaa@7BFE@

U ^ w C z ( σ e 2 ) = l w C z ( σ e 2 ) / σ e 2 = 1 2 i w i j < k s ( i ) w j k | i ( 1 σ e 2 + z i j k 2 2 σ e 4 ) = 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFysPYR0xH8viFfea0dXdHaVhbvd9G8Grpq0xc9vqpq 0xbba9q8WqFfeaY=biNkVc=He9pgeu0dXdar=Jir=JbbG8Fq0xfr=x frVcFf0dmeaabaqaciaacaGaaeqabaWaaeaaeaaakeaaceWGvbGbaK aadaWgaaWcbaGaam4DaiaadoeacaWG6baabeaakmaabmaabaGaeq4W dm3aa0baaSqaaiaadwgaaeaacaaIYaaaaaGccaGLOaGaayzkaaGaey ypa0ZaaSGbaeaacqGHciITcaWGSbWaaSbaaSqaaiaadEhacaWGdbGa amOEaaqabaGcdaqadaqaaiabeo8aZnaaDaaaleaacaWGLbaabaGaaG OmaaaaaOGaayjkaiaawMcaaaqaaiabgkGi2kabeo8aZnaaDaaaleaa caWGLbaabaGaaGOmaaaaaaGccqGH9aqpdaWcaaqaaiaaigdaaeaaca aIYaaaamaaqafabaGaam4DamaaBaaaleaacaWGPbaabeaakmaaqafa baGaam4DamaaBaaaleaadaabcaqaaiaadQgacaWGRbaacaGLiWoaca WGPbaabeaakmaabmaabaGaeyOeI0YaaSaaaeaacaaIXaaabaGaeq4W dm3aa0baaSqaaiaadwgaaeaacaaIYaaaaaaakiabgUcaRmaalaaaba GaamOEamaaDaaaleaacaWGPbGaamOAaiaadUgaaeaacaaIYaaaaaGc baGaaGOmaiabeo8aZnaaDaaaleaacaWGLbaabaGaaGinaaaaaaaaki aawIcacaGLPaaacqGH9aqpcaaIWaaaleaacaWGQbGaeyipaWJaam4A aiabgIGiolaadohacaGGOaGaamyAaiaacMcaaeqaniabggHiLdaale aacaWGPbaabeqdcqGHris5aaaa@7854@

pour obtenir les estimateurs de la vraisemblance composite pondérée (VCP) μ ^ w C , σ ^ v w C 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFysPYR0xH8viFfea0dXdHaVhbvd9G8Grpq0xc9vqpq 0xbba9q8WqFfeaY=biNkVc=He9pgeu0dXdar=Jir=JbbG8Fq0xfr=x frVcFf0dmeaabaqaciaacaGaaeqabaWaaeaaeaaakeaacuaH8oqBga qcamaaBaaaleaacaWG3bGaam4qaaqabaGccaGGSaGafq4WdmNbaKaa daqhaaWcbaGaamODaiaadEhacaWGdbaabaGaaGOmaaaaaaa@3ECC@ et σ ^ e w C 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFysPYR0xH8viFfea0dXdHaVhbvd9G8Grpq0xc9vqpq 0xbba9q8WqFfeaY=biNkVc=He9pgeu0dXdar=Jir=JbbG8Fq0xfr=x frVcFf0dmeaabaqaciaacaGaaeqabaWaaeaaeaaakeaacuaHdpWCga qcamaaDaaaleaacaWGLbGaam4DaiaadoeaaeaacaaIYaaaaaaa@3A4C@ . Les estimateurs VCP sont identiques aux estimateurs (3.9) à (3.11) obtenus par l'approche des équations d'estimation pondérées de la section 3.

Nous nous penchons maintenant sur le modèle de régression linéaire à erreurs emboîtées (3.13). Mentionnons pour commencer que y i j ~ N ( x i j T β , σ 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFysPYR0xH8viFfea0dXdHaVhbvd9G8Grpq0xc9vqpq 0xbba9q8WqFfeaY=biNkVc=He9pgeu0dXdar=Jir=JbbG8Fq0xfr=x frVcFf0dmeaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWG5bWaaS baaSqaaiaadMgacaWGQbaabeaakiaac6hacaWGobWaaeWaaeaacaWH 4bWaa0baaSqaaiaadMgacaWGQbaabaGaamivaaaakiaahk7acaGGSa GaaGjbVlaaykW7cqaHdpWCdaahaaWcbeqaaiaaikdaaaaakiaawIca caGLPaaaaaa@46FB@ , où σ 2 = σ v 2 + σ e 2 , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFysPYR0xH8viFfea0dXdHaVhbvd9G8Grpq0xc9vqpq 0xbba9q8WqFfeaY=biNkVc=He9pgeu0dXdar=Jir=JbbG8Fq0xfr=x frVcFf0dmeaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHdpWCda ahaaWcbeqaaiaaikdaaaGccqGH9aqpcqaHdpWCdaqhaaWcbaGaamOD aaqaaiaaikdaaaGccqGHRaWkcqaHdpWCdaqhaaWcbaGaamyzaaqaai aaikdaaaGccaGGSaaaaa@4181@ et z i j k = y i j y i k ~ N { ( x i j x i k ) T β , 2 σ e 2 } . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFysPYR0xH8viFfea0dXdHaVhbvd9G8Grpq0xc9vqpq 0xbba9q8WqFfeaY=biNkVc=He9pgeu0dXdar=Jir=JbbG8Fq0xfr=x frVcFf0dmeaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWG6bWaaS baaSqaaiaadMgacaWGQbGaam4AaaqabaGccqGH9aqpcaWG5bWaaSba aSqaaiaadMgacaWGQbaabeaakiabgkHiTiaadMhadaWgaaWcbaGaam yAaiaadUgaaeqaaOGaaiOFaiaad6eadaGadaqaamaabmaabaGaaCiE amaaBaaaleaacaWGPbGaamOAaaqabaGccqGHsislcaWH4bWaaSbaaS qaaiaadMgacaWGRbaabeaaaOGaayjkaiaawMcaamaaCaaaleqabaGa amivaaaakiaahk7acaGGSaGaaGjbVlaaykW7caaIYaGaeq4Wdm3aa0 baaSqaaiaadwgaaeaacaaIYaaaaaGccaGL7bGaayzFaaGaaiOlaaaa @58C3@ Il s'ensuit que les équations de score composite pondérées sont données par

U ^ w C y 1 ( β , σ 2 ) = l w C y ( β , σ 2 ) / β = i s w i j s ( i ) w j | i x i j ( y i j x i j T β ) = 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFysPYR0xH8viFfea0dXdHaVhbvd9G8Grpq0xc9vqpq 0xbba9q8WqFfeaY=biNkVc=He9pgeu0dXdar=Jir=JbbG8Fq0xfr=x frVcFf0dmeaabaqaciaacaGaaeqabaWaaeaaeaaakqaaeeqaaiqahw fagaqcamaaBaaaleaacaWG3bGaam4qaiaadMhacaaIXaaabeaakiaa cIcacaWHYoGaaiilaiabeo8aZnaaCaaaleqabaGaaGOmaaaakiaacM cacqGH9aqpdaWcgaqaaiabgkGi2kaadYgadaWgaaWcbaGaam4Daiaa doeacaWG5baabeaakmaabmaabaGaaCOSdiaacYcacaaMe8Uaeq4Wdm 3aaWbaaSqabeaacaaIYaaaaaGccaGLOaGaayzkaaaabaGaeyOaIyRa aCOSdaaaaeaacqGH9aqpdaaeqbqaaiaadEhadaWgaaWcbaGaamyAaa qabaaabaGaamyAaiabgIGiolaadohaaeqaniabggHiLdGcdaaeqbqa aiaadEhadaWgaaWcbaWaaqGaaeaacaWGQbaacaGLiWoacaWGPbaabe aakiaahIhadaWgaaWcbaGaamyAaiaadQgaaeqaaOWaaeWaaeaacaWG 5bWaaSbaaSqaaiaadMgacaWGQbaabeaakiabgkHiTiaahIhadaqhaa WcbaGaamyAaiaadQgaaeaacaWGubaaaOGaaCOSdaGaayjkaiaawMca aiabg2da9iaahcdaaSqaaiaadQgacqGHiiIZcaWGZbGaaiikaiaadM gacaGGPaaabeqdcqGHris5aaaaaa@74FA@

U ^ w C y 2 ( β , σ 2 ) = l w C y ( β , σ 2 ) / σ 2 = 1 2 i s w i j s ( i ) w j | i [ 1 σ 2 ( y i j x i j T β ) 2 σ 4 ] = 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFysPYR0xH8viFfea0dXdHaVhbvd9G8Grpq0xc9vqpq 0xbba9q8WqFfeaY=biNkVc=He9pgeu0dXdar=Jir=JbbG8Fq0xfr=x frVcFf0dmeaabaqaciaacaGaaeqabaWaaeaaeaaakqaaeeqaaiqahw fagaqcamaaBaaaleaacaWG3bGaam4qaiaadMhacaaIYaaabeaakiaa cIcacaWHYoGaaiilaiabeo8aZnaaCaaaleqabaGaaGOmaaaakiaacM cacqGH9aqpdaWcgaqaaiabgkGi2kaadYgadaWgaaWcbaGaam4Daiaa doeacaWG5baabeaakmaabmaabaGaaCOSdiaacYcacaaMe8Uaeq4Wdm 3aaWbaaSqabeaacaaIYaaaaaGccaGLOaGaayzkaaaabaGaeyOaIyRa eq4Wdm3aaWbaaSqabeaacaaIYaaaaaaaaOqaaiabg2da9iabgkHiTm aalaaabaGaaGymaaqaaiaaikdaaaWaaabuaeaacaWG3bWaaSbaaSqa aiaadMgaaeqaaOWaaabuaeaacaWG3bWaaSbaaSqaamaaeiaabaGaam OAaaGaayjcSdGaamyAaaqabaGcdaWadaqaamaalaaabaGaaGymaaqa aiabeo8aZnaaCaaaleqabaGaaGOmaaaaaaGccqGHsisldaWcaaqaam aabmaabaGaamyEamaaBaaaleaacaWGPbGaamOAaaqabaGccqGHsisl caWH4bWaa0baaSqaaiaadMgacaWGQbaabaGaamivaaaakiaahk7aai aawIcacaGLPaaadaahaaWcbeqaaiaaikdaaaaakeaacqaHdpWCdaah aaWcbeqaaiaaisdaaaaaaaGccaGLBbGaayzxaaGaeyypa0JaaGimaa WcbaGaamOAaiabgIGiolaadohacaGGOaGaamyAaiaacMcaaeqaniab ggHiLdaaleaacaWGPbGaeyicI4Saam4Caaqab0GaeyyeIuoaaaaa@7FFA@

et

U ^ w C z ( σ e 2 ) = l w C z ( σ e 2 ) / σ e 2 = 1 2 i s w i j < k s ( i ) w j k | i { 1 σ e 2 [ z i j k ( x i j x i k ) T β ] 2 2 σ e 4 } = 0. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFysPYR0xH8viFfea0dXdHaVhbvd9G8Grpq0xc9vqpq 0xbba9q8WqFfeaY=biNkVc=He9pgeu0dXdar=Jir=JbbG8Fq0xfr=x frVcFf0dmeaabaqaciaacaGaaeqabaWaaeaaeaaakqaaeeqaaiqahw fagaqcamaaBaaaleaacaWG3bGaam4qaiaadQhaaeqaaOGaaiikaiab eo8aZnaaDaaaleaacaWGLbaabaGaaGOmaaaakiaacMcacqGH9aqpda WcgaqaaiabgkGi2kaadYgadaWgaaWcbaGaam4DaiaadoeacaWG6baa beaakmaabmaabaGaeq4Wdm3aa0baaSqaaiaadwgaaeaacaaIYaaaaa GccaGLOaGaayzkaaaabaGaeyOaIyRaeq4Wdm3aa0baaSqaaiaadwga aeaacaaIYaaaaaaaaOqaaiabg2da9iabgkHiTmaalaaabaGaaGymaa qaaiaaikdaaaWaaabuaeaacaWG3bWaaSbaaSqaaiaadMgaaeqaaOWa aabuaeaacaWG3bWaaSbaaSqaamaaeiaabaGaamOAaiaadUgaaiaawI a7aiaadMgaaeqaaOWaaiWaaeaadaWcaaqaaiaaigdaaeaacqaHdpWC daqhaaWcbaGaamyzaaqaaiaaikdaaaaaaOGaeyOeI0YaaSaaaeaada WadaqaaiaadQhadaWgaaWcbaGaamyAaiaadQgacaWGRbaabeaakiab gkHiTmaabmaabaGaaCiEamaaBaaaleaacaWGPbGaamOAaaqabaGccq GHsislcaWH4bWaaSbaaSqaaiaadMgacaWGRbaabeaaaOGaayjkaiaa wMcaamaaCaaaleqabaGaamivaaaakiaahk7aaiaawUfacaGLDbaada ahaaWcbeqaaiaaikdaaaaakeaacaaIYaGaeq4Wdm3aa0baaSqaaiaa dwgaaeaacaaI0aaaaaaaaOGaay5Eaiaaw2haaiabg2da9iaaicdaca GGUaaaleaacaWGQbGaeyipaWJaam4AaiabgIGiolaadohacaGGOaGa amyAaiaacMcaaeqaniabggHiLdaaleaacaWGPbGaeyicI4Saam4Caa qab0GaeyyeIuoaaaaa@8A15@

Les estimateurs VCP résultants de β MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFysPYR0xH8viFfea0dXdHaVhbvd9G8Grpq0xc9vqpq 0xbba9q8WqFfeaY=biNkVc=He9pgeu0dXdar=Jir=JbbG8Fq0xfr=x frVcFf0dmeaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWHYoaaaa@3620@ , σ v 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFysPYR0xH8viFfea0dXdHaVhbvd9G8Grpq0xc9vqpq 0xbba9q8WqFfeaY=biNkVc=He9pgeu0dXdar=Jir=JbbG8Fq0xfr=x frVcFf0dmeaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHdpWCda qhaaWcbaGaamODaaqaaiaaikdaaaaaaa@3889@ et σ e 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFysPYR0xH8viFfea0dXdHaVhbvd9G8Grpq0xc9vqpq 0xbba9q8WqFfeaY=biNkVc=He9pgeu0dXdar=Jir=JbbG8Fq0xfr=x frVcFf0dmeaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHdpWCda qhaaWcbaGaamyzaaqaaiaaikdaaaaaaa@3878@ sont donnés par

β ^ w C = ( i s j s ( i ) w i j x i j x i j T ) 1 ( i s j s ( i ) w i j x i j y i j ) , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFysPYR0xH8viFfea0dXdHaVhbvd9G8Grpq0xc9vqpq 0xbba9q8WqFfeaY=biNkVc=He9pgeu0dXdar=Jir=JbbG8Fq0xfr=x frVcFf0dmeaabaqaciaacaGaaeqabaWaaeaaeaaakeaaceWHYoGbaK aadaWgaaWcbaGaam4DaiaadoeaaeqaaOGaeyypa0ZaaeWaaeaadaae qbqaamaaqafabaGaam4DamaaBaaaleaacaWGPbGaamOAaaqabaGcca WH4bWaaSbaaSqaaiaadMgacaWGQbaabeaakiaahIhadaqhaaWcbaGa amyAaiaadQgaaeaacaWGubaaaaqaaiaadQgacqGHiiIZcaWGZbGaai ikaiaadMgacaGGPaaabeqdcqGHris5aaWcbaGaamyAaiabgIGiolaa dohaaeqaniabggHiLdaakiaawIcacaGLPaaadaahaaWcbeqaaiabgk HiTiaaigdaaaGcdaqadaqaamaaqafabaWaaabuaeaacaWG3bWaaSba aSqaaiaadMgacaWGQbaabeaakiaahIhadaWgaaWcbaGaamyAaiaadQ gaaeqaaOGaamyEamaaBaaaleaacaWGPbGaamOAaaqabaaabaGaamOA aiabgIGiolaadohacaGGOaGaamyAaiaacMcaaeqaniabggHiLdaale aacaWGPbGaeyicI4Saam4Caaqab0GaeyyeIuoaaOGaayjkaiaawMca aabaaaaaaaaapeGaaiilaaaa@6CE3@

σ ^ w C 2 = i s j s ( i ) w i j ( y i j x i j T β ^ w C ) 2 / i s j s ( i ) w i j , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFysPYR0xH8viFfea0dXdHaVhbvd9G8Grpq0xc9vqpq 0xbba9q8WqFfeaY=biNkVc=He9pgeu0dXdar=Jir=JbbG8Fq0xfr=x frVcFf0dmeaabaqaciaacaGaaeqabaWaaeaaeaaakeaacuaHdpWCga qcamaaDaaaleaacaWG3bGaam4qaaqaaiaaikdaaaGccqGH9aqpdaWc gaqaamaaqafabaWaaabuaeaacaWG3bWaaSbaaSqaaiaadMgacaWGQb aabeaakmaabmaabaGaamyEamaaBaaaleaacaWGPbGaamOAaaqabaGc cqGHsislcaWH4bWaa0baaSqaaiaadMgacaWGQbaabaGaamivaaaaki qahk7agaqcamaaBaaaleaacaWG3bGaam4qaaqabaaakiaawIcacaGL PaaadaahaaWcbeqaaiaaikdaaaaabaGaamOAaiabgIGiolaadohaca GGOaGaamyAaiaacMcaaeqaniabggHiLdaaleaacaWGPbGaeyicI4Sa am4Caaqab0GaeyyeIuoakiaaysW7caaMc8oabaGaaGPaVpaaqafaba WaaabuaeaacaWG3bWaaSbaaSqaaiaadMgacaWGQbaabeaaaeaacaWG QbGaeyicI4Saam4CaiaacIcacaWGPbGaaiykaaqab0GaeyyeIuoaaS qaaiaadMgacqGHiiIZcaWGZbaabeqdcqGHris5aaaakiaacYcaaaa@6E56@

et

σ ^ e w C 2 = i s w i j < k s ( i ) w j k | i [ z i j k ( x i j x i k ) T β ^ w C ] 2 / ( 2 i s w i j < k s ( i ) w j k | i ) . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFysPYR0xH8viFfea0dXdHaVhbvd9G8Grpq0xc9vqpq 0xbba9q8WqFfeaY=biNkVc=He9pgeu0dXdar=Jir=JbbG8Fq0xfr=x frVcFf0dmeaabaqaciaacaGaaeqabaWaaeaaeaaakeaacuaHdpWCga qcamaaDaaaleaacaWGLbGaam4DaiaadoeaaeaacaaIYaaaaOGaeyyp a0JaaGPaVpaalyaabaWaaabuaeaacaWG3bWaaSbaaSqaaiaadMgaae qaaOWaaabuaeaacaWG3bWaaSbaaSqaamaaeiaabaGaamOAaiaadUga aiaawIa7aiaadMgaaeqaaaqaaiaadQgacqGH8aapcaWGRbGaeyicI4 Saam4CaiaacIcacaWGPbGaaiykaaqab0GaeyyeIuoaaSqaaiaadMga cqGHiiIZcaWGZbaabeqdcqGHris5aOWaamWaaeaacaWG6bWaaSbaaS qaaiaadMgacaWGQbGaam4AaaqabaGccqGHsisldaqadaqaaiaahIha daWgaaWcbaGaamyAaiaadQgaaeqaaOGaeyOeI0IaaCiEamaaBaaale aacaWGPbGaam4AaaqabaaakiaawIcacaGLPaaadaahaaWcbeqaaiaa dsfaaaGcceWHYoGbaKaadaWgaaWcbaGaam4DaiaadoeaaeqaaaGcca GLBbGaayzxaaWaaWbaaSqabeaacaaIYaaaaaGcbaWaaeWaaeaacaaI YaWaaabuaeaacaWG3bWaaSbaaSqaaiaadMgaaeqaaaqaaiaadMgacq GHiiIZcaWGZbaabeqdcqGHris5aOWaaabuaeaacaWG3bWaaSbaaSqa amaaeiaabaGaamOAaiaadUgaaiaawIa7aiaadMgaaeqaaaqaaiaadQ gacqGH8aapcaWGRbGaeyicI4Saam4CaiaacIcacaWGPbGaaiykaaqa b0GaeyyeIuoaaOGaayjkaiaawMcaaaaacaGGUaaaaa@82B3@

L'estimateur de σ v 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFysPYR0xH8viFfea0dXdHaVhbvd9G8Grpq0xc9vqpq 0xbba9q8WqFfeaY=biNkVc=He9pgeu0dXdar=Jir=JbbG8Fq0xfr=x frVcFf0dmeaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHdpWCda qhaaWcbaGaamODaaqaaiaaikdaaaaaaa@3889@ est donné par σ ^ v w C 2 = σ ^ w C 2 σ ^ e w C 2 . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFysPYR0xH8viFfea0dXdHaVhbvd9G8Grpq0xc9vqpq 0xbba9q8WqFfeaY=biNkVc=He9pgeu0dXdar=Jir=JbbG8Fq0xfr=x frVcFf0dmeaabaqaciaacaGaaeqabaWaaeaaeaaakeaacuaHdpWCga qcamaaDaaaleaacaWG2bGaam4DaiaadoeaaeaacaaIYaaaaOGaeyyp a0Jafq4WdmNbaKaadaqhaaWcbaGaam4DaiaadoeaaeaacaaIYaaaaO GaeyOeI0Iafq4WdmNbaKaadaqhaaWcbaGaamyzaiaadEhacaWGdbaa baGaaGOmaaaakiaac6caaaa@470A@ De nouveau, les estimateurs VCP β ^ W C MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFysPYR0xH8viFfea0dXdHaVhbvd9G8Grpq0xc9vqpq 0xbba9q8WqFfeaY=biNkVc=He9pgeu0dXdar=Jir=JbbG8Fq0xfr=x frVcFf0dmeaabaqaciaacaGaaeqabaWaaeaaeaaakeaaceWHYoGbaK aadaWgaaWcbaGaam4vaiaadoeaaeqaaaaa@3800@ , σ ^ v W C 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFysPYR0xH8viFfea0dXdHaVhbvd9G8Grpq0xc9vqpq 0xbba9q8WqFfeaY=biNkVc=He9pgeu0dXdar=Jir=JbbG8Fq0xfr=x frVcFf0dmeaabaqaciaacaGaaeqabaWaaeaaeaaakeaacuaHdpWCga qcamaaDaaaleaacaWG2bGaam4vaiaadoeaaeaacaaIYaaaaaaa@3A3C@ et σ ^ e W C 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFysPYR0xH8viFfea0dXdHaVhbvd9G8Grpq0xc9vqpq 0xbba9q8WqFfeaY=biNkVc=He9pgeu0dXdar=Jir=JbbG8Fq0xfr=x frVcFf0dmeaabaqaciaacaGaaeqabaWaaeaaeaaakeaacuaHdpWCga qcamaaDaaaleaacaWGLbGaam4vaiaadoeaaeaacaaIYaaaaaaa@3A2B@ sont identiques aux estimateurs (3.17) à (3.19) obtenus par l'approche des équations d'estimation pondérées de la section 3.

L'approche de la vraisemblance composite susmentionnée, fondée sur y i j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFysPYR0xH8viFfea0dXdHaVhbvd9G8Grpq0xc9vqpq 0xbba9q8WqFfeaY=biNkVc=He9pgeu0dXdar=Jir=JbbG8Fq0xfr=x frVcFf0dmeaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWG5bWaaS baaSqaaiaadMgacaWGQbaabeaaaaa@37E9@ et z i j k = y i j y i k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFysPYR0xH8viFfea0dXdHaVhbvd9G8Grpq0xc9vqpq 0xbba9q8WqFfeaY=biNkVc=He9pgeu0dXdar=Jir=JbbG8Fq0xfr=x frVcFf0dmeaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWG6bWaaS baaSqaaiaadMgacaWGQbGaam4AaaqabaGccqGH9aqpcaWG5bWaaSba aSqaaiaadMgacaWGQbaabeaakiabgkHiTiaadMhadaWgaaWcbaGaam yAaiaadUgaaeqaaaaa@40F0@ , n'est pas applicable au modèle à deux niveaux linéaire donné par (2.4), parce que le vecteur de paramètres, θ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFysPYR0xH8viFfea0dXdHaVhbvd9G8Grpq0xc9vqpq 0xbba9q8WqFfeaY=biNkVc=He9pgeu0dXdar=Jir=JbbG8Fq0xfr=x frVcFf0dmeaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWH4oaaaa@3626@ , n'est pas identifiable sous la vraisemblance composite obtenue à partir des y i j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFysPYR0xH8viFfea0dXdHaVhbvd9G8Grpq0xc9vqpq 0xbba9q8WqFfeaY=biNkVc=He9pgeu0dXdar=Jir=JbbG8Fq0xfr=x frVcFf0dmeaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWG5bWaaS baaSqaaiaadMgacaWGQbaabeaaaaa@37E9@ et z i j k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFysPYR0xH8viFfea0dXdHaVhbvd9G8Grpq0xc9vqpq 0xbba9q8WqFfeaY=biNkVc=He9pgeu0dXdar=Jir=JbbG8Fq0xfr=x frVcFf0dmeaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWG6bWaaS baaSqaaiaadMgacaWGQbGaam4Aaaqabaaaaa@38DA@ . Nous devons faire appel à la méthode par paire pour traiter le modèle (2.4).

Marginalement, ( y i j , y i k ) T MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFysPYR0xH8viFfea0dXdHaVhbvd9G8Grpq0xc9vqpq 0xbba9q8WqFfeaY=biNkVc=He9pgeu0dXdar=Jir=JbbG8Fq0xfr=x frVcFf0dmeaabaqaciaacaGaaeqabaWaaeaaeaaakeaadaqadaqaai aadMhadaWgaaWcbaGaamyAaiaadQgaaeqaaOGaaiilaiaadMhadaWg aaWcbaGaamyAaiaadUgaaeqaaaGccaGLOaGaayzkaaWaaWbaaSqabe aacaWGubaaaaaa@3E43@ suit une loi normale bivariée de moyennes x i j T β MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFysPYR0xH8viFfea0dXdHaVhbvd9G8Grpq0xc9vqpq 0xbba9q8WqFfeaY=biNkVc=He9pgeu0dXdar=Jir=JbbG8Fq0xfr=x frVcFf0dmeaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWH4bWaa0 baaSqaaiaadMgacaWGQbaabaGaamivaaaakiaahk7aaaa@3A0E@ et x i k T β MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFysPYR0xH8viFfea0dXdHaVhbvd9G8Grpq0xc9vqpq 0xbba9q8WqFfeaY=biNkVc=He9pgeu0dXdar=Jir=JbbG8Fq0xfr=x frVcFf0dmeaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWH4bWaa0 baaSqaaiaadMgacaWGRbaabaGaamivaaaakiaahk7aaaa@3A0F@ et de matrice de covariance 2 × 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFysPYR0xH8viFfea0dXdHaVhbvd9G8Grpq0xc9vqpq 0xbba9q8WqFfeaY=biNkVc=He9pgeu0dXdar=Jir=JbbG8Fq0xfr=x frVcFf0dmeaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaaIYaGaey 41aqRaaGOmaaaa@3871@

Σ i ( j k ) = [ σ e 2 + x i j T Σ v x i j x i j T Σ v x i k x i k T Σ v x i j σ e 2 + x i k T Σ v x i k ] . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFysPYR0xH8viFfea0dXdHaVhbvd9G8Grpq0xc9vqpq 0xbba9q8WqFfeaY=biNkVc=He9pgeu0dXdar=Jir=JbbG8Fq0xfr=x frVcFf0dmeaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWHJoWaaS baaSqaaiaadMgacaGGOaGaamOAaiaadUgacaGGPaaabeaakiabg2da 9maadmaabaqbaeqabiGaaaqaaiabeo8aZnaaDaaaleaacaWGLbaaba GaaGOmaaaakiabgUcaRiaahIhadaqhaaWcbaGaamyAaiaadQgaaeaa caWGubaaaOGaaC4OdmaaBaaaleaacaWG2baabeaakiaahIhadaWgaa WcbaGaamyAaiaadQgaaeqaaaGcbaGaaCiEamaaDaaaleaacaWGPbGa amOAaaqaaiaadsfaaaGccaWHJoWaaSbaaSqaaiaadAhaaeqaaOGaaC iEamaaBaaaleaacaWGPbGaam4AaaqabaaakeaacaWH4bWaa0baaSqa aiaadMgacaWGRbaabaGaamivaaaakiaaho6adaWgaaWcbaGaamODaa qabaGccaWH4bWaaSbaaSqaaiaadMgacaWGQbaabeaaaOqaaiabeo8a ZnaaDaaaleaacaWGLbaabaGaaGOmaaaakiabgUcaRiaahIhadaqhaa WcbaGaamyAaiaadUgaaeaacaWGubaaaOGaaC4OdmaaBaaaleaacaWG 2baabeaakiaahIhadaWgaaWcbaGaamyAaiaadUgaaeqaaaaaaOGaay 5waiaaw2faaabaaaaaaaaapeGaaiOlaaaa@6CD6@

Maintenant, il découle de (4.3) que les équations de score composite pondérées sont données par

β : U ^ w C β = i s w i j < k s ( i ) w j k | i X i ( j k ) T Σ i ( j k ) 1 ( y i ( j k ) X i ( j k ) T β ) = 0        ( 4.4 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFysPYR0xH8viFfea0dXdHaVhbvd9G8Grpq0xc9vqpq 0xbba9q8WqFfeaY=biNkVc=He9pgeu0dXdar=Jir=JbbG8Fq0xfr=x frVcFf0dmeaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWHYoGaai OoaiaaxMaaceWHvbGbaKaadaWgaaWcbaGaam4DaiaadoeacaWHYoaa beaakiabg2da9maaqafabaGaam4DamaaBaaaleaacaWGPbaabeaakm aaqafabaGaam4DamaaBaaaleaadaabcaqaaiaadQgacaWGRbaacaGL iWoacaWGPbaabeaakiaahIfadaqhaaWcbaGaamyAaiaacIcacaWGQb Gaam4AaiaacMcaaeaacaWGubaaaOGaaC4OdmaaDaaaleaacaWGPbGa aiikaiaadQgacaWGRbGaaiykaaqaaiabgkHiTiaaigdaaaGcdaqada qaaiaahMhadaWgaaWcbaGaamyAaiaacIcacaWGQbGaam4AaiaacMca aeqaaOGaeyOeI0IaaCiwamaaDaaaleaacaWGPbGaaiikaiaadQgaca WGRbGaaiykaaqaaiaadsfaaaGccaWHYoaacaGLOaGaayzkaaaaleaa caWGQbGaeyipaWJaam4AaiabgIGiolaadohacaGGOaGaamyAaiaacM caaeqaniabggHiLdaaleaacaWGPbGaeyicI4Saam4Caaqab0Gaeyye Iuoakiabg2da9iaahcdacaWLjaGaaCzcamaabmaabaaeaaaaaaaaa8 qacaaI0aGaaiOlaiaaisdaa8aacaGLOaGaayzkaaaaaa@771D@

et

τ : U ^ w C l = 1 2 i s w i j < k s ( i ) w j k | i [ ( y i ( j k ) X i ( j k ) T β ) T Σ i ( j k ) 1 Σ i ( j k ) τ l Σ i ( j k ) 1 ( y i ( j k ) X i ( j k ) T β )        ( 4.5 ) tr ( Σ i ( j k ) 1 Σ i ( j k ) τ l ) ] = 0 , l = 1 , ... , p ( p + 1 ) / 2 + 1 = P MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFysPYR0xH8viFfea0dXdHaVhbvd9G8Grpq0xc9vqpq 0xbba9q8WqFfeaY=biNkVc=He9pgeu0dXdar=Jir=JbbG8Fq0xfr=x frVcFf0dmeaabaqaciaacaGaaeqabaWaaeaaeaaakqaabeqaaiaahs 8acaGG6aGaaCzcaiqahwfagaqcamaaBaaaleaacaWG3bGaam4qaiaa dYgaaeqaaOGaeyypa0JaaGPaVlaaykW7caaMc8+aaSaaaeaacaaIXa aabaGaaGOmaaaadaaeqbqaaiaadEhadaWgaaWcbaGaamyAaaqabaGc daaeqbqaaiaadEhadaWgaaWcbaWaaqGaaeaacaWGQbGaam4AaaGaay jcSdGaamyAaaqabaGcdaWabaqaamaabmaabaGaaCyEamaaBaaaleaa caWGPbGaaiikaiaadQgacaWGRbGaaiykaaqabaGccqGHsislcaWHyb Waa0baaSqaaiaadMgacaGGOaGaamOAaiaadUgacaGGPaaabaGaamiv aaaakiaahk7aaiaawIcacaGLPaaadaahaaWcbeqaaiaadsfaaaGcca WHJoWaa0baaSqaaiaadMgacaGGOaGaamOAaiaadUgacaGGPaaabaGa eyOeI0IaaGymaaaakmaalaaabaGaeyOaIyRaaC4OdmaaBaaaleaaca WGPbGaaiikaiaadQgacaWGRbGaaiykaaqabaaakeaacqGHciITcqaH epaDdaWgaaWcbaGaamiBaaqabaaaaOGaaC4OdmaaDaaaleaacaWGPb GaaiikaiaadQgacaWGRbGaaiykaaqaaiabgkHiTiaaigdaaaGcdaqa daqaaiaahMhadaWgaaWcbaGaamyAaiaacIcacaWGQbGaam4AaiaacM caaeqaaOGaeyOeI0IaaCiwamaaDaaaleaacaWGPbGaaiikaiaadQga caWGRbGaaiykaaqaaiaadsfaaaGccaWHYoaacaGLOaGaayzkaaaaca GLBbaaaSqaaiaadQgacqGH8aapcaWGRbGaeyicI4Saam4CaiaacIca caWGPbGaaiykaaqab0GaeyyeIuoaaSqaaiaadMgacqGHiiIZcaWGZb aabeqdcqGHris5aOGaaCzcaiaaxMaadaqadaqaaabaaaaaaaaapeGa aGinaiaac6cacaaI1aaapaGaayjkaiaawMcaaaqaamaadiaabaGaaC zcaiaaxMaacqGHsislcaqG0bGaaeOCamaabmaabaGaaC4OdmaaDaaa leaacaWGPbGaaiikaiaadQgacaWGRbGaaiykaaqaaiabgkHiTiaaig daaaGcdaWcaaqaaiabgkGi2kaaho6adaWgaaWcbaGaamyAaiaacIca caWGQbGaam4AaiaacMcaaeqaaaGcbaGaeyOaIyRaeqiXdq3aaSbaaS qaaiaadYgaaeqaaaaaaOGaayjkaiaawMcaaaGaayzxaaGaeyypa0Ja aCimaiaacYcacaaMc8UaaCzcaiaaxMaacaWLjaGaamiBaiabg2da9i aaigdacaGGSaGaaiOlaiaac6cacaGGUaGaaiilamaalyaabaGaamiC aiaacIcacaWGWbGaey4kaSIaaGymaiaacMcaaeaacaaIYaaaaiabgU caRiaaigdacqGH9aqpcaWGqbaaaaa@C754@

X i ( j k ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFysPYR0xH8viFfea0dXdHaVhbvd9G8Grpq0xc9vqpq 0xbba9q8WqFfeaY=biNkVc=He9pgeu0dXdar=Jir=JbbG8Fq0xfr=x frVcFf0dmeaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWHybWaaS baaSqaaiaadMgacaGGOaGaamOAaiaadUgacaGGPaaabeaaaaa@3A15@ est la matrice de dimensions 2 × p MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFysPYR0xH8viFfea0dXdHaVhbvd9G8Grpq0xc9vqpq 0xbba9q8WqFfeaY=biNkVc=He9pgeu0dXdar=Jir=JbbG8Fq0xfr=x frVcFf0dmeaabaqaciaacaGaaeqabaWaaeaaeaaakeaajaaycaaIYa Gaey41aqRaamiCaaaa@3913@ contenant les lignes x i j T MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFysPYR0xH8viFfea0dXdHaVhbvd9G8Grpq0xc9vqpq 0xbba9q8WqFfeaY=biNkVc=He9pgeu0dXdar=Jir=JbbG8Fq0xfr=x frVcFf0dmeaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWH4bWaa0 baaSqaaiaadMgacaWGQbaabaGaamivaaaaaaa@38C6@ et x i k T MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFysPYR0xH8viFfea0dXdHaVhbvd9G8Grpq0xc9vqpq 0xbba9q8WqFfeaY=biNkVc=He9pgeu0dXdar=Jir=JbbG8Fq0xfr=x frVcFf0dmeaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWH4bWaa0 baaSqaaiaadMgacaWGRbaabaGaamivaaaaaaa@38C7@ , y i ( j k ) = ( y i j , y i k ) T MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFysPYR0xH8viFfea0dXdHaVhbvd9G8Grpq0xc9vqpq 0xbba9q8WqFfeaY=biNkVc=He9pgeu0dXdar=Jir=JbbG8Fq0xfr=x frVcFf0dmeaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWH5bWaaS baaSqaaiaadMgacaGGOaGaamOAaiaadUgacaGGPaaabeaakiabg2da 9maabmaabaGaamyEamaaBaaaleaacaWGPbGaamOAaaqabaGccaGGSa GaamyEamaaBaaaleaacaWGPbGaam4AaaqabaaakiaawIcacaGLPaaa daahaaWcbeqaaiaadsfaaaaaaa@44A8@ , et τ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFysPYR0xH8viFfea0dXdHaVhbvd9G8Grpq0xc9vqpq 0xbba9q8WqFfeaY=biNkVc=He9pgeu0dXdar=Jir=JbbG8Fq0xfr=x frVcFf0dmeaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWHepaaaa@3632@ est le vecteur de dimension P contenant les éléments τ 1 = σ e 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFysPYR0xH8viFfea0dXdHaVhbvd9G8Grpq0xc9vqpq 0xbba9q8WqFfeaY=biNkVc=He9pgeu0dXdar=Jir=JbbG8Fq0xfr=x frVcFf0dmeaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHepaDda WgaaWcbaGaaGymaaqabaGccqGH9aqpcqaHdpWCdaqhaaWcbaGaamyz aaqaaiaaikdaaaaaaa@3C34@ et les p ( p + 1 ) / 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFysPYR0xH8viFfea0dXdHaVhbvd9G8Grpq0xc9vqpq 0xbba9q8WqFfeaY=biNkVc=He9pgeu0dXdar=Jir=JbbG8Fq0xfr=x frVcFf0dmeaabaqaciaacaGaaeqabaWaaeaaeaaakeaadaWcgaqcaa waaiaadchakmaabmaajaaybaGaamiCaiabgUcaRiaaigdaaiaawIca caGLPaaaaeaacaaIYaaaaaaa@3BA0@ éléments distincts de Σ v MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFysPYR0xH8viFfea0dXdHaVhbvd9G8Grpq0xc9vqpq 0xbba9q8WqFfeaY=biNkVc=He9pgeu0dXdar=Jir=JbbG8Fq0xfr=x frVcFf0dmeaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWHJoWaaS baaSqaaiaadAhaaeqaaaaa@3738@ désignés par τ 2 , ... , τ P MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFysPYR0xH8viFfea0dXdHaVhbvd9G8Grpq0xc9vqpq 0xbba9q8WqFfeaY=biNkVc=He9pgeu0dXdar=Jir=JbbG8Fq0xfr=x frVcFf0dmeaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHepaDda WgaaWcbaGaaGOmaaqabaGccaGGSaGaaiOlaiaac6cacaGGUaGaaiil aiabes8a0naaBaaaleaacaWGqbaabeaaaaa@3DD5@ . Nous pouvons résoudre les équations de score composite pondérées (4.4) et (4.5) itérativement en utilisant la méthode de Newton-Raphson ou une autre méthode itérative pour obtenir les estimateurs VCP β ^ w C MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFysPYR0xH8viFfea0dXdHaVhbvd9G8Grpq0xc9vqpq 0xbba9q8WqFfeaY=biNkVc=He9pgeu0dXdar=Jir=JbbG8Fq0xfr=x frVcFf0dmeaabaqaciaacaGaaeqabaWaaeaaeaaakeaaceWHYoGbaK aadaWgaaWcbaGaam4Daiaadoeaaeqaaaaa@3820@ et  τ ^ w C MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFysPYR0xH8viFfea0dXdHaVhbvd9G8Grpq0xc9vqpq 0xbba9q8WqFfeaY=biNkVc=He9pgeu0dXdar=Jir=JbbG8Fq0xfr=x frVcFf0dmeaabaqaciaacaGaaeqabaWaaeaaeaaakeaaceWHepGbaK aadaWgaaWcbaGaam4Daiaadoeaaeqaaaaa@3832@ .

Dans le cas particulier du modèle de régression linéaire à erreurs emboîtées (3.13), les équations de score de recensement, fondées sur la log-vraisemblance de recensement complète l ( θ ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFysPYR0xH8viFfea0dXdHaVhbvd9G8Grpq0xc9vqpq 0xbba9q8WqFfeaY=biNkVc=He9pgeu0dXdar=Jir=JbbG8Fq0xfr=x frVcFf0dmeaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGSbGaai ikaiaahI7acaGGPaaaaa@3870@ donnée par (2.5), peuvent s'écrire sous une forme explicite. Les équations de score pondérées d'échantillon correspondantes ne dépendent que des poids de niveau 1 w j | i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFysPYR0xH8viFfea0dXdHaVhbvd9G8Grpq0xc9vqpq 0xbba9q8WqFfeaY=biNkVc=He9pgeu0dXdar=Jir=JbbG8Fq0xfr=x frVcFf0dmeaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWG3bWaaS baaSqaamaaeiaabaGaamOAaaGaayjcSdGaamyAaaqabaaaaa@397D@ et w j k | i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFysPYR0xH8viFfea0dXdHaVhbvd9G8Grpq0xc9vqpq 0xbba9q8WqFfeaY=biNkVc=He9pgeu0dXdar=Jir=JbbG8Fq0xfr=x frVcFf0dmeaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWG3bWaaS baaSqaamaaeiaabaGaamOAaiaadUgaaiaawIa7aiaadMgaaeqaaaaa @3A6D@ et des poids de niveau 2 w i , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFysPYR0xH8viFfea0dXdHaVhbvd9G8Grpq0xc9vqpq 0xbba9q8WqFfeaY=biNkVc=He9pgeu0dXdar=Jir=JbbG8Fq0xfr=x frVcFf0dmeaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWG3bWaaS baaSqaaiaadMgaaeqaaOGaaiilaaaa@37B2@ comme les équations de score composite pondérées (voir l'annexe). Les estimateurs résultants sont convergents sous le modèle pour θ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFysPYR0xH8viFfea0dXdHaVhbvd9G8Grpq0xc9vqpq 0xbba9q8WqFfeaY=biNkVc=He9pgeu0dXdar=Jir=JbbG8Fq0xfr=x frVcFf0dmeaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWH4oaaaa@3626@ , contrairement aux estimateurs fondés sur la pseudo log-vraisemblance pondérée l w ( θ ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFysPYR0xH8viFfea0dXdHaVhbvd9G8Grpq0xc9vqpq 0xbba9q8WqFfeaY=biNkVc=He9pgeu0dXdar=Jir=JbbG8Fq0xfr=x frVcFf0dmeaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGSbWaaS baaSqaaiaadEhaaeqaaOGaaiikaiaahI7acaGGPaaaaa@39A2@ donnés par (2.7) et (2.8). Cependant, pour des modèles plus complexes, comme les modèles à deux niveaux avec pentes aléatoires, les équations de score pondérées d'échantillon dépendront des probabilités d'inclusion de niveau 1 d'ordres 3 et 4, contrairement aux équations de score composite pondérées (4.3) qui ne dépendent que des probabilités d'inclusion de niveau 1 d'ordres 1 et 2, même pour les modèles multiniveaux complexes. Par conséquent, nous n'avons pas inclus l'approche des équations de score pondérées fondée sur la log-vraisemblance de recensement complète dans l'étude en simulation.

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