Comptes Rendus
no. 12
Statistics/Probability Theory
Test of homogeneity in semiparametric two-sample density ratio models
[Un test de comparaison de lois pour des modèles à rapport de densités semi-paramétriques]

Présenté par : Paul Deheuvels

Amor Keziou1 ; Samuela Leoni-Aubin2
1 LSTA, université Paris 6, 175, rue du Chevaleret, boîte 158, 75013 Paris, France
2 Dipartimento di Scienze Statistiche, Università di Padova, via 8 Febbraio, 2, 35122 Padova, Italy
Comptes Rendus. Mathématique, Volume 340 (2005) no. 12, pp. 905-910.

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.

Reçu le :
Accepté le :
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DOI : 10.1016/j.crma.2005.04.034
Amor Keziou 1 ; Samuela Leoni-Aubin 2

1 LSTA, université Paris 6, 175, rue du Chevaleret, boîte 158, 75013 Paris, France
2 Dipartimento di Scienze Statistiche, Università di Padova, via 8 Febbraio, 2, 35122 Padova, Italy 
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Amor Keziou; Samuela Leoni-Aubin. Test of homogeneity in semiparametric two-sample density ratio models. Comptes Rendus. Mathématique, Volume 340 (2005) no. 12, pp. 905-910. doi : 10.1016/j.crma.2005.04.034. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2005.04.034/

[1] M. Broniatowski 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] T.F. Cox; G. Ferry Robust logistic discrimination, Biometrika, Volume 78 (1991) no. 4, pp. 841-849

[5] K. Fokianos; B. Kedem; J. Qin; D.A. Short A semiparametric approach to the one-way layout, Technometrics, Volume 43 (2001) no. 1, pp. 56-65

[6] V.P. Godambe An optimum property of regular maximum likelihood estimation, Ann. Math. Statist., Volume 31 (1960), pp. 1208-1211

[7] M. Hollander; D.A. Wolfe Nonparametric Statistical Methods, Wiley, New York, 1999

[8] R. Kay; S. Little Transformations of the explanatory variables in the logistic regression model for binary data, Biometrika, Volume 74 (1987) no. 3, pp. 495-501

[9] A. Keziou Dual representation of ϕ-divergences and applications, C. R. Acad. Sci. Paris, Ser. I, Volume 336 (2003) no. 10, pp. 857-862

[10] F. Liese; I. Vajda Convex Statistical Distances, Teubner-Texte Math., vol. 95, Teubner, Leipzig, 1987

[11] D. Morales; L. Pardo Some approximations to power functions of ϕ-divergences tests in parametric models, Test, Volume 10 (2001) no. 2, pp. 249-269

[12] A.B. Owen Empirical Likelihood, Chapman and Hall, New York, 2001

[13] A.B. Owen Empirical likelihood ratio confidence regions, Ann. Statist., Volume 18 (1990) no. 1, pp. 90-120

[14] A.B. Owen Empirical likelihood ratio confidence intervals for a single functional, Biometrika, Volume 75 (1988) no. 2, pp. 237-249

[15] J. Qin Inferences for case-control and semiparametric two-sample density ratio models, Biometrika, Volume 85 (1998) no. 3, pp. 619-630

[16] R.T. Rockafellar Convex Analysis, Princeton University Press, Princeton, NJ, 1970

[17] F. Zou; J.P. Fine; B.S. Yandell On empirical likelihood for a semiparametric mixture model, Biometrika, Volume 89 (2002) no. 1, pp. 61-75

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