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

Section 3. Approche proposée

Pour tout α ( 0, 1 ) , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySdeMaey icI48aaeWaaeaacaaIWaGaaGilaiaaysW7caaIXaaacaGLOaGaayzk aaGaaiilaaaa@3F0B@ le α e MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySde2aaW baaSqabeaacaqGLbaaaaaa@38AB@ quantile d’une distribution F MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOraaaa@36C2@ se définit comme

ξ α = inf { u : F ( u ) α } . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOVdG3aaS baaSqaaiabeg7aHbqabaGccaaI9aGaaeyAaiaab6gacaqGMbWaaiWa aeaacaWG1bGaaGjbVlaaiQdacaaMe8UaamOramaabmaabaGaamyDaa GaayjkaiaawMcaaiabgwMiZkabeg7aHbGaay5Eaiaaw2haaiaai6ca aaa@4B8F@

Si F ^ ( u ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaja WaaeWaaeaacaWG1baacaGLOaGaayzkaaaaaa@3955@ est une estimation de F ( u ) , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOramaabm aabaGaamyDaaGaayjkaiaawMcaaiaacYcaaaa@39F5@ son α MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySdegaaa@3796@ -quantile est naturellement estimé à l’aide de

ξ ^ α = inf { u : F ^ ( u ) α } . ( 3.1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqOVdGNbaK aadaWgaaWcbaGaeqySdegabeaakiaai2daciGGPbGaaiOBaiaacAga daGadaqaaiaadwhacaaMe8UaaGOoaiaaysW7ceWGgbGbaKaadaqada qaaiaadwhaaiaawIcacaGLPaaacqGHLjYScqaHXoqyaiaawUhacaGL 9baacaaIUaGaaGzbVlaaywW7caaMf8UaaGzbVlaaywW7caGGOaGaaG 4maiaac6cacaaIXaGaaiykaaaa@56FD@

Selon l’hypothèse de distribution pour ϵ i j , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWefv3ySLgznf gDOfdaryqr1ngBPrginfgDObYtUvgaiuaacqWF1pG8daWgaaWcbaGa amyAaiaadQgaaeqaaOGaaiilaaaa@44B3@ nous avons

P ( y i j u ) = E { P ( ε i j u m 0 ( x i j ) v i | x i j , v i ) } = E { G i ( u m 0 ( x i j ) v i ) } . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqbaeaabiGaaa qaaiaadcfadaqadaqaaiaadMhadaWgaaWcbaGaamyAaiaadQgaaeqa aOGaeyizImQaamyDaaGaayjkaiaawMcaaaqaaiaai2datuuDJXwAK1 uy0HMmaeHbfv3ySLgzG0uy0HgiuD3BaGqbaiab=ri8fnaacmaabaGa amiuamaabmaabaGaeqyTdu2aaSbaaSqaaiaadMgacaWGQbaabeaaki abgsMiJkaadwhacqGHsislcaWGTbWaaSbaaSqaaiaaicdaaeqaaOWa aeWaaeaacaWG4bWaaSbaaSqaaiaadMgacaWGQbaabeaaaOGaayjkai aawMcaaiabgkHiTmaaeiaabaGaamODamaaBaaaleaacaWGPbaabeaa kiaaykW7aiaawIa7aiaaykW7caWG4bWaaSbaaSqaaiaadMgacaWGQb aabeaakiaaiYcacaaMe8UaamODamaaBaaaleaacaWGPbaabeaaaOGa ayjkaiaawMcaaaGaay5Eaiaaw2haaaqaaaqaaiaai2dacqWFecFrda GadaqaaiaadEeadaWgaaWcbaGaamyAaaqabaGcdaqadaqaaiaadwha cqGHsislcaWGTbWaaSbaaSqaaiaaicdaaeqaaOWaaeWaaeaacaWG4b WaaSbaaSqaaiaadMgacaWGQbaabeaaaOGaayjkaiaawMcaaiabgkHi TiaadAhadaWgaaWcbaGaamyAaaqabaaakiaawIcacaGLPaaaaiaawU hacaGL9baacaGGUaaaaaaa@8124@

Par conséquent, nous obtenons la distribution de la population du i e MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAamaaCa aaleqabaGaaeyzaaaaaaa@37FA@ petit domaine comme suit

F i ( u ) = N i 1 j = 1 N i G i ( u m 0 ( x i j ) v i ) . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOramaaBa aaleaacaWGPbaabeaakmaabmaabaGaamyDaaGaayjkaiaawMcaaiaa i2dacaWGobWaa0baaSqaaiaadMgaaeaacqGHsislcaaIXaaaaOWaaa bCaeqaleaacaWGQbGaaGypaiaaigdaaeaacaWGobWaaSbaaWqaaiaa dMgaaeqaaaqdcqGHris5aOGaam4ramaaBaaaleaacaWGPbaabeaakm aabmaabaGaamyDaiabgkHiTiaad2gadaWgaaWcbaGaaGimaaqabaGc daqadaqaaiaadIhadaWgaaWcbaGaamyAaiaadQgaaeqaaaGccaGLOa GaayzkaaGaeyOeI0IaamODamaaBaaaleaacaWGPbaabeaaaOGaayjk aiaawMcaaiaai6caaaa@551A@

Une fois que G i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4ramaaBa aaleaacaWGPbaabeaaaaa@37DD@ et m 0 ( ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBamaaBa aaleaacaaIWaaabeaakmaabmaabaGaeyyXICnacaGLOaGaayzkaaaa aa@3BAC@ seront correctement estimés, il en ira de même des quantiles sur petits domaines.

Nous suivons l’idée de la vraisemblance empirique de Chen et Liu (2018) pour l’estimation de G i ( ) . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4ramaaBa aaleaacaWGPbaabeaakmaabmaabaGaeyyXICnacaGLOaGaayzkaaGa aiOlaaaa@3C6C@ Supposons que les valeurs de ε i j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyTdu2aaS baaSqaaiaadMgacaWGQbaabeaaaaa@39A7@ dans l’échantillon sont connues. Prenons un candidat G 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4ramaaBa aaleaacaaIWaaabeaaaaa@37A9@ sous la forme suivante

G 0 ( u ) = i , j p i j I ( ε i j u ) , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4ramaaBa aaleaacaaIWaaabeaakmaabmaabaGaamyDaaGaayjkaiaawMcaaiaa i2dadaaeqbqaaiaadchadaWgaaWcbaGaamyAaiaadQgaaeqaaaqaai aadMgacaaISaGaaGPaVlaadQgaaeqaniabggHiLdGccaWGjbWaaeWa aeaacqaH1oqzdaWgaaWcbaGaamyAaiaadQgaaeqaaOGaeyizImQaam yDaaGaayjkaiaawMcaaiaaiYcaaaa@4DB0@

I ( ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysamaabm aabaGaeyyXICnacaGLOaGaayzkaaaaaa@3A98@ est une fonction indicatrice et i,j = i=0 m   j=1 n i  . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaabeaeqale aacaWGPbGaaGilaiaaykW7caWGQbaabeqdcqGHris5aOGaaGypamaa qadabeWcbaGaamyAaiaai2dacaaIWaaabaGaamyBaaqdcqGHris5aO GaaeiiamaaqadabeWcbaGaamOAaiaai2dacaaIXaaabaGaamOBamaa BaaameaacaWGPbaabeaaa0GaeyyeIuoakiaabccacaGGUaaaaa@4ABC@ Par conséquent, nous avons p i j = d G 0 ( ε i j ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCamaaBa aaleaacaWGPbGaamOAaaqabaGccaaI9aGaamizaiaadEeadaWgaaWc baGaaGimaaqabaGcdaqadaqaaiabew7aLnaaBaaaleaacaWGPbGaam OAaaqabaaakiaawIcacaGLPaaaaaa@41AE@ et, selon le MRD, d G i ( ε s t ) = p s t exp { θ i q ( ε s t ) } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamizaiaadE eadaWgaaWcbaGaamyAaaqabaGcdaqadaqaaiabew7aLnaaBaaaleaa caWGZbGaamiDaaqabaaakiaawIcacaGLPaaacaaI9aGaamiCamaaBa aaleaacaWGZbGaamiDaaqabaGcciGGLbGaaiiEaiaacchadaGadaqa aiaahI7adaqhaaWcbaGaamyAaaqaaKqzGfGamai2gkdiIcaakiaahg hadaqadaqaaiabew7aLnaaBaaaleaacaWGZbGaamiDaaqabaaakiaa wIcacaGLPaaaaiaawUhacaGL9baaaaa@537F@ pour i = 0, 1, , m MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAaiaai2 dacaaIWaGaaGilaiaaysW7caaIXaGaaGilaiaaysW7cqWIMaYscaaI SaGaaGjbVlaad2gaaaa@41FE@ ce qui implique ceci

G i ( u ) = s , t p s t exp { θ i q ( ε s t ) } I ( ε s t u ) . ( 3.2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4ramaaBa aaleaacaWGPbaabeaakmaabmaabaGaamyDaaGaayjkaiaawMcaaiaa i2dadaaeqbqaaiaadchadaWgaaWcbaGaam4Caiaadshaaeqaaaqaai aadohacaaISaGaaGPaVlaadshaaeqaniabggHiLdGcciGGLbGaaiiE aiaacchadaGadaqaaiaahI7adaqhaaWcbaGaamyAaaqaaKqzGfGama i2gkdiIcaakiaahghadaqadaqaaiabew7aLnaaBaaaleaacaWGZbGa amiDaaqabaaakiaawIcacaGLPaaaaiaawUhacaGL9baacaWGjbWaae WaaeaacqaH1oqzdaWgaaWcbaGaam4CaiaadshaaeqaaOGaeyizImQa amyDaaGaayjkaiaawMcaaiaai6cacaaMf8UaaGzbVlaaywW7caaMf8 UaaGzbVlaacIcacaaIZaGaaiOlaiaaikdacaGGPaaaaa@6AE1@

Selon Owen (2001), nous obtenons la fonction de vraisemblance empirique

L n ( G 0 , G 1 , , G m ) = i , j d G i ( ε i j ) = { i , j p i j } exp [ i , j { θ i q ( ε i j ) } ] , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamitamaaBa aaleaacaWGUbaabeaakmaabmaabaGaam4ramaaBaaaleaacaaIWaaa beaakiaaiYcacaaMe8Uaam4ramaaBaaaleaacaaIXaaabeaakiaaiY cacaaMe8UaeSOjGSKaaGilaiaaysW7caWGhbWaaSbaaSqaaiaad2ga aeqaaaGccaGLOaGaayzkaaGaaGypamaarafabaGaamizaiaadEeada WgaaWcbaGaamyAaaqabaGcdaqadaqaaiabew7aLnaaBaaaleaacaWG PbGaamOAaaqabaaakiaawIcacaGLPaaaaSqaaiaadMgacaaISaGaaG PaVlaadQgaaeqaniabg+GivdGccaaI9aWaaiWaaeaadaqeqbqaaiaa dchadaWgaaWcbaGaamyAaiaadQgaaeqaaaqaaiaadMgacaaISaGaaG PaVlaadQgaaeqaniabg+GivdaakiaawUhacaGL9baaciGGLbGaaiiE aiaacchadaWadaqaamaaqafabeWcbaGaamyAaiaaiYcacaaMc8Uaam OAaaqab0GaeyyeIuoakmaacmaabaGaaGjcVlaaysW7caWH4oWaa0ba aSqaaiaadMgaaeaajugybiadaITHYaIOaaGccaWHXbWaaeWaaeaacq aH1oqzdaWgaaWcbaGaamyAaiaadQgaaeqaaaGccaGLOaGaayzkaaaa caGL7bGaayzFaaaacaGLBbGaayzxaaGaaGilaaaa@7F86@

où le paramètre θ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCiUdaaa@373B@ et p i j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCamaaBa aaleaacaWGPbGaamOAaaqabaaaaa@38F5@ satisfait p i j 0 , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCamaaBa aaleaacaWGPbGaamOAaaqabaGccqGHLjYScaaIWaGaaiilaaaa@3C2F@ et où s = 0, 1, , m , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4Caiaai2 dacaaIWaGaaGilaiaaysW7caaIXaGaaGilaiaaysW7cqWIMaYscaaI SaGaaGjbVlaad2gacaGGSaaaaa@42B8@

i , j p i j exp { θ s q ( ε i j ) } = 1. ( 3.3 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaabuaeaaca WGWbWaaSbaaSqaaiaadMgacaWGQbaabeaaaeaacaWGPbGaaGilaiaa ykW7caWGQbaabeqdcqGHris5aOGaciyzaiaacIhacaGGWbWaaiWaae aacaWH4oWaa0baaSqaaiaadohaaeaajugybiadaITHYaIOaaGccaWH XbWaaeWaaeaacqaH1oqzdaWgaaWcbaGaamyAaiaadQgaaeqaaaGcca GLOaGaayzkaaaacaGL7bGaayzFaaGaaGypaiaaigdacaaIUaGaaGzb VlaaywW7caaMf8UaaGzbVlaaywW7caGGOaGaaG4maiaac6cacaaIZa Gaaiykaaaa@5E24@

À noter que nous avons utilisé la convention θ 0 = 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCiUdmaaBa aaleaacaaIWaaabeaakiaai2dacaaIWaaaaa@39AC@ pour simplifier la présentation. Parce que G 1 , , G m MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4ramaaBa aaleaacaaIXaaabeaakiaaiYcacaaMe8UaeSOjGSKaaGilaiaaysW7 caWGhbWaaSbaaSqaaiaad2gaaeqaaaaa@3F46@ sont entièrement déterminés par θ = ( θ 1 , , θ m ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabCiUdyaafa GaaGypamaabmaabaGaaCiUdmaaDaaaleaacaaIXaaabaqcLbwacWaG yBOmGikaaOGaaGilaiaaysW7cqWIMaYscaaISaGaaGjbVlaahI7ada qhaaWcbaGaamyBaaqaaKqzGfGamai2gkdiIcaaaOGaayjkaiaawMca aaaa@4B40@ et G 0 , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4ramaaBa aaleaacaaIWaaabeaakiaacYcaaaa@3863@ nous écrivons le log-vraisemblance empirique comme ceci

l n ( θ , G 0 ) = i , j log ( p i j ) + i j θ i q ( ε i j ) . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeS4eHW2aaS baaSqaaiaad6gaaeqaaOWaaeWaaeaacaWH4oGaaGilaiaadEeadaWg aaWcbaGaaGimaaqabaaakiaawIcacaGLPaaacaaI9aWaaabuaeqale aacaWGPbGaaGilaiaaykW7caWGQbaabeqdcqGHris5aOGaciiBaiaa c+gacaGGNbWaaeWaaeaacaWGWbWaaSbaaSqaaiaadMgacaWGQbaabe aaaOGaayjkaiaawMcaaiabgUcaRmaaqafabaGaaCiUdmaaDaaaleaa caWGPbaabaqcLbwacWaGyBOmGikaaaWcbaGaamyAaiaadQgaaeqani abggHiLdGccaWHXbWaaeWaaeaacqaH1oqzdaWgaaWcbaGaamyAaiaa dQgaaeqaaaGccaGLOaGaayzkaaGaaGOlaaaa@5DF2@

En optimisant l ( θ , G 0 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeS4eHW2aae WaaeaacaWH4oGaaGilaiaaysW7caWGhbWaaSbaaSqaaiaaicdaaeqa aaGccaGLOaGaayzkaaaaaa@3DF4@ en ce qui concerne G 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4ramaaBa aaleaacaaIWaaabeaaaaa@37A9@ selon les résultats des contraintes (3.3) dans les probabilités ajustées

p ^ i j = n 1 { 1 + s = 1 m λ s [ exp { θ s q ( ε i j ) } 1 ] } 1 ( 3.4 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmiCayaaja WaaSbaaSqaaiaadMgacaWGQbaabeaakiaai2dacaWGUbWaaWbaaSqa beaacqGHsislcaaIXaaaaOWaaiWaaeaacaaIXaGaey4kaSYaaabCae aacqaH7oaBdaWgaaWcbaGaam4CaaqabaaabaGaam4Caiaai2dacaaI XaaabaGaamyBaaqdcqGHris5aOWaamWaaeaaciGGLbGaaiiEaiaacc hadaGadaqaaiaahI7adaqhaaWcbaGaam4CaaqaaKqzGfGamai2gkdi IcaakiaahghacaaIOaGaeqyTdu2aaSbaaSqaaiaadMgacaWGQbaabe aakiaaiMcaaiaawUhacaGL9baacqGHsislcaaIXaaacaGLBbGaayzx aaaacaGL7bGaayzFaaWaaWbaaSqabeaacqGHsislcaaIXaaaaOGaaG zbVlaaywW7caaMf8UaaGzbVlaaywW7caGGOaGaaG4maiaac6cacaaI 0aGaaiykaaaa@6B06@

et le log-VE du profil

l n ( θ ) = i , j log { 1 + s = 1 m λ s [ exp { θ s q ( ε i j ) } 1 ] } + i , j θ i q ( ε i j ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeS4eHW2aaS baaSqaaiaad6gaaeqaaOWaaeWaaeaacaWH4oaacaGLOaGaayzkaaGa aGypaiabgkHiTmaaqafabeWcbaGaamyAaiaaiYcacaaMc8UaamOAaa qab0GaeyyeIuoakiGacYgacaGGVbGaai4zamaacmaabaGaaGymaiab gUcaRmaaqahabaGaeq4UdW2aaSbaaSqaaiaadohaaeqaaaqaaiaado hacaaI9aGaaGymaaqaaiaad2gaa0GaeyyeIuoakmaadmaabaGaciyz aiaacIhacaGGWbWaaiWaaeaacaWH4oWaa0baaSqaaiaadohaaeaaju gybiadaITHYaIOaaGccaWHXbWaaeWaaeaacqaH1oqzdaWgaaWcbaGa amyAaiaadQgaaeqaaaGccaGLOaGaayzkaaaacaGL7bGaayzFaaGaey OeI0IaaGymaaGaay5waiaaw2faaaGaay5Eaiaaw2haaiabgUcaRmaa qafabaGaaCiUdmaaDaaaleaacaWGPbaabaqcLbwacWaGyBOmGikaaa WcbaGaamyAaiaaiYcacaaMc8UaamOAaaqab0GaeyyeIuoakiaahgha daqadaqaaiabew7aLnaaBaaaleaacaWGPbGaamOAaaqabaaakiaawI cacaGLPaaaaaa@7ABD@

( λ 1 , , λ m ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaacq aH7oaBdaWgaaWcbaGaaGymaaqabaGccaaISaGaaGjbVlablAciljaa iYcacaaMe8Uaeq4UdW2aaSbaaSqaaiaad2gaaeqaaaGccaGLOaGaay zkaaaaaa@42A9@ est la solution de

s,t exp{ θ i q( ε st ) }1 1+ l=1 m   λ l [ exp{ θ l q( ε st ) }1 ] =0. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaabuaeqale aacaWGZbGaaGilaiaaykW7caWG0baabeqdcqGHris5aOGaaGjbVpaa laaabaGaciyzaiaacIhacaGGWbWaaiWaaeaacaWH4oWaa0baaSqaai aadMgaaeaajugybiadaITHYaIOaaGccaWHXbWaaeWaaeaacqaH1oqz daWgaaWcbaGaam4CaiaadshaaeqaaaGccaGLOaGaayzkaaaacaGL7b GaayzFaaGaeyOeI0IaaGymaaqaaiaaigdacqGHRaWkdaaeWaqaaiaa bccacqaH7oaBdaWgaaWcbaGaamiBaaqabaaabaGaamiBaiaai2daca aIXaaabaGaamyBaaqdcqGHris5aOWaamWaaeaaciGGLbGaaiiEaiaa cchadaGadaqaaiaahI7adaqhaaWcbaGaamiBaaqaaKqzGfGamai2gk diIcaakiaahghadaqadaqaaiabew7aLnaaBaaaleaacaWGZbGaamiD aaqabaaakiaawIcacaGLPaaaaiaawUhacaGL9baacqGHsislcaaIXa aacaGLBbGaayzxaaaaaiaai2dacaaIWaGaaGOlaaaa@72D2@

Puisque les valeurs de ε i j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyTdu2aaS baaSqaaiaadMgacaWGQbaabeaaaaa@39A7@ ne sont pas disponibles, nous les remplaçons par les résidus obtenus à l’aide du modèle d’ajustement (2.1) selon l’hypothèse (2.2) :

ε ^ i j = y i j m ^ 0 ( x i j ; β ^ , γ ^ ) v ^ i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqyTduMbaK aadaWgaaWcbaGaamyAaiaadQgaaeqaaOGaaGypaiaadMhadaWgaaWc baGaamyAaiaadQgaaeqaaOGaeyOeI0IabmyBayaajaWaaSbaaSqaai aaicdaaeqaaOWaaeWaaeaacaWG4bWaaSbaaSqaaiaadMgacaWGQbaa beaakiaaiUdacaaMe8UabCOSdyaajaGaaGilaiaaysW7ceWHZoGbaK aaaiaawIcacaGLPaaacqGHsislceWG2bGbaKaadaWgaaWcbaGaamyA aaqabaaaaa@4F54@

m ^ 0 ( x ; β ^ , γ ^ ) = β ^ 0 + β ^ 1 x + + β ^ p x p + k = 1 K γ ^ k ( x κ k ) + p . ( 3.5 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmyBayaaja WaaSbaaSqaaiaaicdaaeqaaOGaaGikaiaadIhacaaI7aGaaGjbVlqa hk7agaqcaiaaiYcacaaMe8UabC4SdyaajaGaaGykaiaai2dacuaHYo GygaqcamaaBaaaleaacaaIWaaabeaakiabgUcaRiqbek7aIzaajaWa aSbaaSqaaiaaigdaaeqaaOGaamiEaiabgUcaRiablAciljabgUcaRi qbek7aIzaajaWaaSbaaSqaaiaadchaaeqaaOGaamiEamaaCaaaleqa baGaamiCaaaakiabgUcaRmaaqahabaGafq4SdCMbaKaadaWgaaWcba Gaam4AaaqabaaabaGaam4Aaiaai2dacaaIXaaabaGaam4saaqdcqGH ris5aOWaaeWaaeaacaWG4bGaeyOeI0IaeqOUdS2aaSbaaSqaaiaadU gaaeqaaaGccaGLOaGaayzkaaWaa0baaSqaaiabgUcaRaqaaiaadcha aaGccaaIUaGaaGzbVlaaywW7caaMf8UaaGzbVlaaywW7caGGOaGaaG 4maiaac6cacaaI1aGaaiykaaaa@6EE6@

Supposons que l ^ n ( θ ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafS4eHWMbaK aadaWgaaWcbaGaamOBaaqabaGcdaqadaqaaiaahI7aaiaawIcacaGL Paaaaaa@3B2E@ est la fonction du log-VE l ˜ n ( θ ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafS4eHWMbaG aadaWgaaWcbaGaamOBaaqabaGcdaqadaqaaiaahI7aaiaawIcacaGL Paaaaaa@3B2D@ après avoir remplacé ε i j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyTdu2aaS baaSqaaiaadMgacaWGQbaabeaaaaa@39A7@ par ε ^ i j . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqyTduMbaK aadaWgaaWcbaGaamyAaiaadQgaaeqaaOGaaiOlaaaa@3A73@ Nous définissons l’estimateur VE maximum de θ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCiUdaaa@373B@ à l’aide de θ ^ = argmax l ^ n ( θ ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabCiUdyaaja GaaGypaiaabggacaqGYbGaae4zaiaab2gacaqGHbGaaeiEaiaayIW7 cuWItecBgaqcamaaBaaaleaacaWGUbaabeaakmaabmaabaGaaCiUda GaayjkaiaawMcaaaaa@446C@ et nous estimons G i ( u ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4ramaaBa aaleaacaWGPbaabeaakmaabmaabaGaamyDaaGaayjkaiaawMcaaaaa @3A6A@ comme ceci

G ˜ i ( u ) = s , t p ^ s t exp { θ ^ i q ( ε ^ s t ) } I ( ε ^ s t u ) ( 3.6 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabm4rayaaia WaaSbaaSqaaiaadMgaaeqaaOWaaeWaaeaacaWG1baacaGLOaGaayzk aaGaaGypamaaqafabaGabmiCayaajaWaaSbaaSqaaiaadohacaWG0b aabeaaaeaacaWGZbGaaGilaiaaykW7caWG0baabeqdcqGHris5aOGa ciyzaiaacIhacaGGWbWaaiWaaeaaceWH4oGbaKaadaqhaaWcbaGaam yAaaqaaKqzGfGamai2gkdiIcaakiaahghadaqadaqaaiqbew7aLzaa jaWaaSbaaSqaaiaadohacaWG0baabeaaaOGaayjkaiaawMcaaaGaay 5Eaiaaw2haaiaadMeadaqadaqaaiqbew7aLzaajaWaaSbaaSqaaiaa dohacaWG0baabeaakiabgsMiJkaadwhaaiaawIcacaGLPaaacaaMf8 UaaGzbVlaaywW7caaMf8UaaGzbVlaacIcacaaIZaGaaiOlaiaaiAda caGGPaaaaa@6A7C@

p ^ s t = n 1 { 1 + l = 1 m ( n l / n ) [ exp { θ l q ( ε ^ s t ) } 1 ] } 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmiCayaaja WaaSbaaSqaaiaadohacaWG0baabeaakiaai2dacaWGUbWaaWbaaSqa beaacqGHsislcaaIXaaaaOWaaiWaaeaacaaIXaGaey4kaSYaaabCae qaleaacaWGSbGaaGypaiaaigdaaeaacaWGTbaaniabggHiLdGcdaqa daqaamaalyaabaGaamOBamaaBaaaleaacaWGSbaabeaaaOqaaiaad6 gaaaaacaGLOaGaayzkaaWaamWaaeaaciGGLbGaaiiEaiaacchadaGa daqaaiaahI7adaqhaaWcbaGaamiBaaqaaKqzGfGamai2gkdiIcaaki aahghadaqadaqaaiqbew7aLzaajaWaaSbaaSqaaiaadohacaWG0baa beaaaOGaayjkaiaawMcaaaGaay5Eaiaaw2haaiabgkHiTiaaigdaai aawUfacaGLDbaaaiaawUhacaGL9baadaahaaWcbeqaaiabgkHiTiaa igdaaaaaaa@61DE@

et θ ^ 0 = 0. MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabCiUdyaaja WaaSbaaSqaaiaaicdaaeqaaOGaaGypaiaaicdacaGGUaaaaa@3A6E@ La routine R drmdel MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeizaiaabk hacaqGTbGaaeizaiaabwgacaqGSbaaaa@3B80@ peut servir à calculer θ ^ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabCiUdyaaja aaaa@374B@ et p ^ i j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmiCayaaja WaaSbaaSqaaiaadMgacaWGQbaabeaaaaa@3905@ qui offre 11 possibilités de fonction de base q ( u ) . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCyCamaabm aabaGaamyDaaGaayjkaiaawMcaaiaac6caaaa@3A26@

Parce que G ˜ i ( u ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabm4rayaaia WaaSbaaSqaaiaadMgaaeqaaOWaaeWaaeaacaWG1baacaGLOaGaayzk aaaaaa@3A79@ est discret, la distribution lissée des noyaux G ^ i ( u ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabm4rayaaja WaaSbaaSqaaiaadMgaaeqaaOWaaeWaaeaacaWG1baacaGLOaGaayzk aaaaaa@3A7A@ suivante donne une meilleure estimation des quantiles :

G ^ i ( u ) = j = 1 n i w ^ i j Φ ( ε ^ i j u b ) , ( 3.7 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabm4rayaaja WaaSbaaSqaaiaadMgaaeqaaOWaaeWaaeaacaWG1baacaGLOaGaayzk aaGaaGypamaaqahabaGabm4DayaajaWaaSbaaSqaaiaadMgacaWGQb aabeaaaeaacaWGQbGaaGypaiaaigdaaeaacaWGUbWaaSbaaWqaaiaa dMgaaeqaaaqdcqGHris5aOGaeuOPdy0aaeWaaeaadaWcaaqaaiqbew 7aLzaajaWaaSbaaSqaaiaadMgacaWGQbaabeaakiabgkHiTiaadwha aeaacaWGIbaaaaGaayjkaiaawMcaaiaaiYcacaaMf8UaaGzbVlaayw W7caaMf8UaaGzbVlaacIcacaaIZaGaaiOlaiaaiEdacaGGPaaaaa@5AC5@

où les poids sont choisis comme w ^ i j = G ˜ i ( ε ^ i j ) G ˜ i ( ε ^ i j ) , b MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabm4Dayaaja WaaSbaaSqaaiaadMgacaWGQbaabeaakiaai2daceWGhbGbaGaadaWg aaWcbaGaamyAaaqabaGcdaqadaqaaiqbew7aLzaajaWaaSbaaSqaai aadMgacaWGQbaabeaaaOGaayjkaiaawMcaaiabgkHiTiqadEeagaac amaaBaaaleaacaWGPbaabeaakmaabmaabaGafqyTduMbaKaadaWgaa WcbaGaamyAaiaadQgaaeqaaOGaeyOeI0cacaGLOaGaayzkaaGaaiil aiaaysW7caWGIbaaaa@4D7F@ est un paramètre de largeur de bande, et Φ ( ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuOPdy0aae WaaeaacqGHflY1aiaawIcacaGLPaaaaaa@3B44@ est la fonction de distribution de la normale type. Comme l’ont proposé Chen et Liu (2013), nous choisissons b=1,06 n 1/5  min{ σ ^ , Q ^ / 1,34 } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOyaiaai2 dacaqGXaGaaeilaiaabcdacaqG2aGaamOBamaaCaaaleqabaGaeyOe I0IaaGymaiaai+cacaaI1aaaaOGaaeiiaiGac2gacaGGPbGaaiOBam aacmaabaGafq4WdmNbaKaacaaISaGaaGjbVpaalyaabaGabmyuayaa jaaabaGaaeymaiaabYcacaqGZaGaaeinaaaaaiaawUhacaGL9baaaa a@4C46@ σ ^ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafq4WdmNbaK aaaaa@37CA@ est l’écart-type de la distribution G ^ i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabm4rayaaja WaaSbaaSqaaiaadMgaaeqaaaaa@37ED@ et Q ^ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmyuayaaja aaaa@36DD@ est son intervalle interquartile.

Dans certaines applications, seules les moyennes de puissances des covariables de la population sont connues et peuvent être utilisées pour établir une inférence statistique. Dans d’autres applications, les covariables de tous les membres de la population sont connues. Cela donne deux estimations possibles des quantiles. Dans le premier cas, nous estimons F i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOramaaBa aaleaacaWGPbaabeaaaaa@37DC@ comme

F ^ i ( a ) ( u ) = n i 1 j = 1 n i G ^ i ( u Y ¯ ^ i { m ^ 0 ( x i j ; β ^ , γ ^ ) m ^ 0 ( x ¯ i ; β ^ , γ ^ ) } ) , ( 3.8 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaja Waa0baaSqaaiaadMgaaeaadaqadaqaaiaadggaaiaawIcacaGLPaaa aaGcdaqadaqaaiaadwhaaiaawIcacaGLPaaacaaI9aGaamOBamaaDa aaleaacaWGPbaabaGaeyOeI0IaaGymaaaakmaaqahabaGabm4rayaa jaWaaSbaaSqaaiaadMgaaeqaaaqaaiaadQgacaaI9aGaaGymaaqaai aad6gadaWgaaadbaGaamyAaaqabaaaniabggHiLdGcdaqadaqaaiaa dwhacqGHsislceWGzbGbaeHbaKaadaWgaaWcbaGaamyAaaqabaGccq GHsisldaGadaqaaiqad2gagaqcamaaBaaaleaacaaIWaaabeaakmaa bmaabaGaamiEamaaBaaaleaacaWGPbGaamOAaaqabaGccaaI7aGaaG jbVlqahk7agaqcaiaaiYcacaaMe8UabC4SdyaajaaacaGLOaGaayzk aaGaeyOeI0IabmyBayaajaWaaSbaaSqaaiaaicdaaeqaaOWaaeWaae aaceWG4bGbaebadaWgaaWcbaGaamyAaaqabaGccaaI7aGaaGjbVlqa hk7agaqcaiaaiYcacaaMe8UabC4SdyaajaaacaGLOaGaayzkaaaaca GL7bGaayzFaaaacaGLOaGaayzkaaGaaGilaiaaywW7caaMf8UaaGzb VlaaywW7caaMf8UaaiikaiaaiodacaGGUaGaaGioaiaacMcaaaa@7A72@

où nous utilisons m ^ 0 ( x ¯ i ; β ^ , γ ^ ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmyBayaaja WaaSbaaSqaaiaaicdaaeqaaOWaaeWaaeaaceWG4bGbaebadaWgaaWc baGaamyAaaqabaGccaaI7aGaaGjbVlqahk7agaqcaiaaiYcacaaMe8 UabC4SdyaajaaacaGLOaGaayzkaaaaaa@42DD@ précisés sous (3.5).

Lorsque les données du recensement sur x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaaaa@36F4@ sont disponibles, nous estimons F i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOramaaBa aaleaacaWGPbaabeaaaaa@37DC@ comme ceci

F ^ i ( b ) ( u ) = N i 1 { j s i I ( y i j u ) + j r i G ^ i ( u m ^ 0 ( x i j ) v ^ i ) } , ( 3.9 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaja Waa0baaSqaaiaadMgaaeaadaqadaqaaiaadkgaaiaawIcacaGLPaaa aaGcdaqadaqaaiaadwhaaiaawIcacaGLPaaacaaI9aGaamOtamaaDa aaleaacaWGPbaabaGaeyOeI0IaaGymaaaakmaacmaabaWaaabuaeaa caWGjbWaaeWaaeaacaWG5bWaaSbaaSqaaiaadMgacaWGQbaabeaaki abgsMiJkaadwhaaiaawIcacaGLPaaaaSqaaiaadQgacqGHiiIZcaWG ZbWaaSbaaWqaaiaadMgaaeqaaaWcbeqdcqGHris5aOGaey4kaSYaaa buaeaaceWGhbGbaKaadaWgaaWcbaGaamyAaaqabaGcdaqadaqaaiaa dwhacqGHsislceWGTbGbaKaadaWgaaWcbaGaaGimaaqabaGcdaqada qaaiaadIhadaWgaaWcbaGaamyAaiaadQgaaeqaaaGccaGLOaGaayzk aaGaeyOeI0IabmODayaajaWaaSbaaSqaaiaadMgaaeqaaaGccaGLOa GaayzkaaaaleaacaWGQbGaeyicI4SaamOCamaaBaaameaacaWGPbaa beaaaSqab0GaeyyeIuoaaOGaay5Eaiaaw2haaiaaiYcacaaMf8UaaG zbVlaaywW7caaMf8UaaGzbVlaacIcacaaIZaGaaiOlaiaaiMdacaGG Paaaaa@7513@

s i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4CamaaBa aaleaacaWGPbaabeaaaaa@3809@ et r i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCamaaBa aaleaacaWGPbaabeaaaaa@3808@ sont des ensembles d’unités observées et non observées dans le petit domaine i . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAaiaac6 caaaa@3797@ Le reste des spécifications est identique à (3.8).

Les estimations proposées ressemblent à celles de Chen et Liu (2018), mais nous utilisons une régression non paramétrique. Parce qu’il est plus facile de recueillir les moyennes de puissances des covariables de la population que les valeurs des covariables de toutes les unités de la population, F ^ i ( a ) ( u ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaja Waa0baaSqaaiaadMgaaeaadaqadaqaaiaadggaaiaawIcacaGLPaaa aaGcdaqadaqaaiaadwhaaiaawIcacaGLPaaaaaa@3CE9@ s’applique de façon plus large que F ^ i ( b ) ( u ) . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaja Waa0baaSqaaiaadMgaaeaadaqadaqaaiaadkgaaiaawIcacaGLPaaa aaGcdaqadaqaaiaadwhaaiaawIcacaGLPaaacaGGUaaaaa@3D9C@ Les calculs sont également plus efficaces. Parce que F ^ i ( b ) ( u ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xc9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbbf9v8Gq0db9qqpm0dXdHqpq0=vr 0=vr0=edbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaja Waa0baaSqaaiaadMgaaeaadaqadaqaaiaadkgaaiaawIcacaGLPaaa aaGcdaqadaqaaiaadwhaaiaawIcacaGLPaaaaaa@3CEA@ utilise les valeurs des covariables de toutes les unités de la population, il devrait donner de meilleurs résultats statistiques quand les deux s’appliquent.


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