[Un test de comparaison de lois pour des modèles à rapport de densités semi-paramétriques]
Nous considérons les problèmes d'estimation et de test à deux échantillon dans des modèles à rapport de densités semi-paramétriques. La vraisemblance empirique pose un problème d'irrégularité sous l'hypothèse nulle d'egalité des deux lois. Nous montrons qu'une forme « duale » de la vraisemblance empirique est bien définie. Un test statistique, basé sur la forme duale de la vraisemblance empirique, est ensuite proposé. Les propriétés asymptotiques de la statistique du test sont étudiées sous l'hypothèse nulle et sous l'hypothèse alternative, et une approximation pour la fonction de puissance est déduite.
We consider estimation and test problems for some semiparametric two-sample density ratio models. The profile empirical likelihood (EL) poses an irregularity problem under the null hypothesis that the laws of the two samples are equal. We show that a ‘dual’ form of the profile EL is well defined even under the null hypothesis. A statistical test, based on the dual form of the EL ratio statistic (ELRS), is then proposed. We give an interpretation for the dual form of the ELRS through ϕ-divergences and ‘duality’ technique. The asymptotic properties of the test statistic are presented both under the null and the alternative hypotheses, and an approximation to the power function is deduced.
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@article{CRMATH_2005__340_12_905_0,
author = {Keziou, Amor and Leoni-Aubin, Samuela},
title = {Test of homogeneity in semiparametric two-sample density ratio models},
journal = {Comptes Rendus. Math\'ematique},
pages = {905--910},
publisher = {Elsevier},
volume = {340},
number = {12},
year = {2005},
doi = {10.1016/j.crma.2005.04.034},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.crma.2005.04.034/}
}
TY - JOUR AU - Keziou, Amor AU - Leoni-Aubin, Samuela TI - Test of homogeneity in semiparametric two-sample density ratio models JO - Comptes Rendus. Mathématique PY - 2005 SP - 905 EP - 910 VL - 340 IS - 12 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2005.04.034/ DO - 10.1016/j.crma.2005.04.034 LA - en ID - CRMATH_2005__340_12_905_0 ER -
%0 Journal Article %A Keziou, Amor %A Leoni-Aubin, Samuela %T Test of homogeneity in semiparametric two-sample density ratio models %J Comptes Rendus. Mathématique %D 2005 %P 905-910 %V 340 %N 12 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2005.04.034/ %R 10.1016/j.crma.2005.04.034 %G en %F CRMATH_2005__340_12_905_0
Keziou, Amor; Leoni-Aubin, Samuela. Test of homogeneity in semiparametric two-sample density ratio models. Comptes Rendus. Mathématique, Tome 340 (2005) no. 12, pp. 905-910. doi : 10.1016/j.crma.2005.04.034. http://www.numdam.org/articles/10.1016/j.crma.2005.04.034/
[1] Estimation of the Kullback–Leibler divergence, Math. Methods Statist., Volume 12 (2004) no. 4, pp. 391-409
[2] M. Broniatowski, A. Keziou, Parametric estimation and testing through divergences, Preprint 2004-1, L.S.T. A – Université Paris 6, 2003
[3] M. Broniatowski, A. Keziou, On the minimization of ϕ-divergences on sets of signed measures, Studia Sci. Math. Hungar. (2005), in press
[4] Robust logistic discrimination, Biometrika, Volume 78 (1991) no. 4, pp. 841-849
[5] A semiparametric approach to the one-way layout, Technometrics, Volume 43 (2001) no. 1, pp. 56-65
[6] An optimum property of regular maximum likelihood estimation, Ann. Math. Statist., Volume 31 (1960), pp. 1208-1211
[7] Nonparametric Statistical Methods, Wiley, New York, 1999
[8] Transformations of the explanatory variables in the logistic regression model for binary data, Biometrika, Volume 74 (1987) no. 3, pp. 495-501
[9] Dual representation of ϕ-divergences and applications, C. R. Acad. Sci. Paris, Ser. I, Volume 336 (2003) no. 10, pp. 857-862
[10] Convex Statistical Distances, Teubner-Texte Math., vol. 95, Teubner, Leipzig, 1987
[11] Some approximations to power functions of ϕ-divergences tests in parametric models, Test, Volume 10 (2001) no. 2, pp. 249-269
[12] Empirical Likelihood, Chapman and Hall, New York, 2001
[13] Empirical likelihood ratio confidence regions, Ann. Statist., Volume 18 (1990) no. 1, pp. 90-120
[14] Empirical likelihood ratio confidence intervals for a single functional, Biometrika, Volume 75 (1988) no. 2, pp. 237-249
[15] Inferences for case-control and semiparametric two-sample density ratio models, Biometrika, Volume 85 (1998) no. 3, pp. 619-630
[16] Convex Analysis, Princeton University Press, Princeton, NJ, 1970
[17] On empirical likelihood for a semiparametric mixture model, Biometrika, Volume 89 (2002) no. 1, pp. 61-75
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