Estimation de quantiles sur petits domaines à l’aide de la régression spline et de la vraisemblance empirique

Section 4. Estimation bootstrap des erreurs quadratiques moyennes

Les estimateurs de quantiles sur petits domaines proposés sont assemblés en suivant de nombreuses étapes intermédiaires. Il est difficile d’évaluer de manière analytique les variances ou l’erreur quadratique moyenne (EQM) de ces estimateurs. Nous suivons d’autres chercheurs (Sinha et Rao (2009), Tzavidis et coll. (2010) et Chen et Liu (2018)) afin d’élaborer une procédure bootstrap comme suit :

Étape 1
Obtenir les estimations β ^ , γ ^ , σ ^ v 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabCOSdyaaja GaaGilaiaaysW7ceWHZoGbaKaacaaISaGaaGjbVlqbeo8aZzaajaWa a0baaSqaaiaadAhaaeaacaaIYaaaaaaa@40D1@ et m ^ 0 ( x , β ^ , γ ^ ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmyBayaaja WaaSbaaSqaaiaaicdaaeqaaOWaaeWaaeaacaWG4bGaaGilaiaaysW7 ceWHYoGbaKaacaaISaGaaGjbVlqaho7agaqcaaGaayjkaiaawMcaaa aa@4192@ à partir du modèle (2.1), et calculer G ^ i ( u ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabm4rayaaja WaaSbaaSqaaiaadMgaaeqaaOWaaeWaaeaacaWG1baacaGLOaGaayzk aaaaaa@3A7A@ comme dans (3.7).
Étape 2
Générer une population bootstrap finie  H * = { y i j * , x i j } , i = 0, , m , j = 1, , N i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamisamaaCa aaleqabaGaaiOkaaaakiaai2dadaGadaqaaiaadMhadaqhaaWcbaGa amyAaiaadQgaaeaacaGGQaaaaOGaaGilaiaaysW7caWG4bWaaSbaaS qaaiaadMgacaWGQbaabeaaaOGaay5Eaiaaw2haaiaaiYcacaaMe8Ua amyAaiaai2dacaaIWaGaaGilaiaaysW7cqWIMaYscaaISaGaaGjbVl aad2gacaaISaGaaGjbVlaadQgacaaI9aGaaGymaiaaiYcacaaMe8Ua eSOjGSKaaGilaiaaysW7caWGobWaaSbaaSqaaiaadMgaaeqaaaaa@5B49@ avec

y i j * = m ^ 0 ( x i j , β ^ , γ ^ ) + v i * + ε i j * , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEamaaDa aaleaacaWGPbGaamOAaaqaaiaacQcaaaGccaaI9aGabmyBayaajaWa aSbaaSqaaiaaicdaaeqaaOWaaeWaaeaacaWG4bWaaSbaaSqaaiaadM gacaWGQbaabeaakiaaiYcacaaMe8UabCOSdyaajaGaaGilaiaaysW7 ceWHZoGbaKaaaiaawIcacaGLPaaacqGHRaWkcaWG2bWaa0baaSqaai aadMgaaeaacaGGQaaaaOGaey4kaSIaeqyTdu2aa0baaSqaaiaadMga caWGQbaabaGaaiOkaaaakiaaiYcaaaa@51DC@

où les résidus bootstrap ε i j * MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyTdu2aa0 baaSqaaiaadMgacaWGQbaabaGaaiOkaaaaaaa@3A56@ sont échantillonnés à partir de la FDC G ^ i ( u ) , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabm4rayaaja WaaSbaaSqaaiaadMgaaeqaaOWaaeWaaeaacaWG1baacaGLOaGaayzk aaGaaiilaaaa@3B2A@ et les v i * MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODamaaDa aaleaacaWGPbaabaGaaiOkaaaaaaa@38BB@ sont générés à partir de N ( 0, σ ^ v 2 ) . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOtamaabm aabaGaaGimaiaaiYcacaaMe8Uafq4WdmNbaKaadaqhaaWcbaGaamOD aaqaaiaaikdaaaaakiaawIcacaGLPaaacaGGUaaaaa@3FC3@
Étape 3
À partir de la population bootstrap H * , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamisamaaCa aaleqabaGaaiOkaaaakiaacYcaaaa@3859@ nous sélectionnons n i * = n i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBamaaDa aaleaacaWGPbaabaGaaiOkaaaakiaai2dacaWGUbWaaSbaaSqaaiaa dMgaaeqaaaaa@3B91@ unités d’échantillonnage dans le petit domaine i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAaaaa@36E5@ par échantillonnage aléatoire simple sans remise, et nous le répétons L MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamitaaaa@36C8@ fois pour obtenir h l * , l = 1, , L . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiAamaaDa aaleaacaWGSbaabaGaaiOkaaaakiaaiYcacaaMe8UaamiBaiaai2da caaIXaGaaGilaiaaysW7cqWIMaYscaaISaGaaGjbVlaadYeacaGGUa aaaa@449B@ Pour chaque échantillon h l * , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiAamaaDa aaleaacaWGSbaabaGaaiOkaaaakiaacYcaaaa@396A@ il faut calculer les estimations F ^ i ( a ) * l ( u ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaja Waa0baaSqaaiaadMgaaeaadaqadaqaaiaadggaaiaawIcacaGLPaaa caGGQaGaamiBaaaakmaabmaabaGaamyDaaGaayjkaiaawMcaaaaa@3E88@ et F ^ i ( b ) * l ( u ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaja Waa0baaSqaaiaadMgaaeaadaqadaqaaiaadkgaaiaawIcacaGLPaaa caGGQaGaamiBaaaakmaabmaabaGaamyDaaGaayjkaiaawMcaaaaa@3E89@ comme dans (3.8) et (3.9) respectivement.
Étape 4
Calculer l’estimateur de l’EQM empirique de τ ^ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiXdqNbaK aaaaa@37CC@ comme ceci

eqm ( τ * ) = L 1 l = 1 L ( τ ^ * l τ * ) 2 , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeyzaiaabg hacaqGTbWaaeWaaeaacqaHepaDdaahaaWcbeqaaiaacQcaaaaakiaa wIcacaGLPaaacaaI9aGaamitamaaCaaaleqabaGaeyOeI0IaaGymaa aakmaaqahabaWaaeWaaeaacuaHepaDgaqcamaaCaaaleqabaGaaiOk aiaadYgaaaGccqGHsislcqaHepaDdaahaaWcbeqaaiaacQcaaaaaki aawIcacaGLPaaadaahaaWcbeqaaiaaikdaaaaabaGaamiBaiaai2da caaIXaaabaGaamitaaqdcqGHris5aOGaaGilaaaa@505B@

τ ^ * l = τ ( F ^ * l ( u ) ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiXdqNbaK aadaahaaWcbeqaaiaacQcacaWGSbaaaOGaaGypaiabes8a0naabmaa baGabmOrayaajaWaaWbaaSqabeaacaGGQaGaamiBaaaakmaabmaaba GaamyDaaGaayjkaiaawMcaaaGaayjkaiaawMcaaaaa@42EB@ désigne une fonction de F ^ ( a ) * l ( u ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaja WaaWbaaSqabeaadaqadaqaaiaadggaaiaawIcacaGLPaaacaGGQaGa amiBaaaakmaabmaabaGaamyDaaGaayjkaiaawMcaaaaa@3D9A@ ou F ^ ( b ) * l MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaja WaaWbaaSqabeaadaqadaqaaiaadkgaaiaawIcacaGLPaaacaGGQaGa amiBaaaaaaa@3B0E@ et τ * = τ ( F * ( u ) ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiXdq3aaW baaSqabeaacaGGQaaaaOGaaGypaiabes8a0naabmaabaGaamOramaa CaaaleqabaGaaiOkaaaakmaabmaabaGaamyDaaGaayjkaiaawMcaaa GaayjkaiaawMcaaaaa@40E9@ F * ( u ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOramaaCa aaleqabaGaaiOkaaaakmaabmaabaGaamyDaaGaayjkaiaawMcaaaaa @3A2A@ est la FDC connue des populations bootstrap.
Étape 5
Répéter les étapes 2 à 4, B fois, et définir l’estimation de l’EQM bootstrap comme ceci

B 1 b = 1 B eqm ( τ * ) b , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOqamaaCa aaleqabaGaeyOeI0IaaGymaaaakmaaqahabaGaaeyzaiaabghacaqG TbWaaeWaaeaacqaHepaDdaahaaWcbeqaaiaacQcaaaaakiaawIcaca GLPaaadaWgaaWcbaGaamOyaaqabaaabaGaamOyaiaai2dacaaIXaaa baGaamOqaaqdcqGHris5aOGaaGilaaaa@46D5@

eqm ( τ * ) b MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeyzaiaabg hacaqGTbWaaeWaaeaacqaHepaDdaahaaWcbeqaaiaacQcaaaaakiaa wIcacaGLPaaadaWgaaWcbaGaamOyaaqabaaaaa@3E09@ est eqm( τ * ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeyzaiaabg hacaqGTbWaaeWaaeaacqaHepaDdaahaaWcbeqaaiaaiQcaaaGccaqG GaaacaGLOaGaayzkaaaaaa@3D9F@ calculée à la b e MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOyamaaCa aaleqabaGaaeyzaaaaaaa@37F3@ répétition.

Le rendement de l’estimateur de l’EQM bootstrap sera examiné et déclaré dans la section portant sur la simulation.


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