The likelihood ratio test for general mixture models with or without structural parameter
ESAIM: Probability and Statistics, Tome 13 (2009), pp. 301-327.

Nous étudions le test du rapport de vraisemblance (TRV) pour des hypothèses sur la mesure mélangeante dans un mélange en présence éventuelle d'un paramètre structurel, et ce dans toutes les situations possibles. Le résultat principal donne la distribution asymptotique du TRV sous des hypothèses qui ne sont pas loin d'être nécessaires. Nous donnons une solution détaillée pour le test d'une simple distribution contre un mélange avec application aux lois gaussiennes, Poisson et binomiales, ainsi que pour le test du nombre de populations dans un mélange fini avec un paramètre structurel.

This paper deals with the likelihood ratio test (LRT) for testing hypotheses on the mixing measure in mixture models with or without structural parameter. The main result gives the asymptotic distribution of the LRT statistics under some conditions that are proved to be almost necessary. A detailed solution is given for two testing problems: the test of a single distribution against any mixture, with application to gaussian, Poisson and binomial distributions; the test of the number of populations in a finite mixture with or without structural parameter.

DOI : 10.1051/ps:2008010
Classification : 62F05, 62F12, 62H10, 62H30
Mots clés : likelihood ratio test, mixture models, number of components, local power, contiguity 
@article{PS_2009__13__301_0,
     author = {Aza{\"\i}s, Jean-Marc and Gassiat, \'Elisabeth and Mercadier, C\'ecile},
     title = {The likelihood ratio test for general mixture models with or without structural parameter},
     journal = {ESAIM: Probability and Statistics},
     pages = {301--327},
     publisher = {EDP-Sciences},
     volume = {13},
     year = {2009},
     doi = {10.1051/ps:2008010},
     mrnumber = {2528086},
     zbl = {1180.62069},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/ps:2008010/}
}
TY  - JOUR
AU  - Azaïs, Jean-Marc
AU  - Gassiat, Élisabeth
AU  - Mercadier, Cécile
TI  - The likelihood ratio test for general mixture models with or without structural parameter
JO  - ESAIM: Probability and Statistics
PY  - 2009
SP  - 301
EP  - 327
VL  - 13
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/ps:2008010/
DO  - 10.1051/ps:2008010
LA  - en
ID  - PS_2009__13__301_0
ER  - 
%0 Journal Article
%A Azaïs, Jean-Marc
%A Gassiat, Élisabeth
%A Mercadier, Cécile
%T The likelihood ratio test for general mixture models with or without structural parameter
%J ESAIM: Probability and Statistics
%D 2009
%P 301-327
%V 13
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/ps:2008010/
%R 10.1051/ps:2008010
%G en
%F PS_2009__13__301_0
Azaïs, Jean-Marc; Gassiat, Élisabeth; Mercadier, Cécile. The likelihood ratio test for general mixture models with or without structural parameter. ESAIM: Probability and Statistics, Tome 13 (2009), pp. 301-327. doi : 10.1051/ps:2008010. http://www.numdam.org/articles/10.1051/ps:2008010/

[1] R.J. Adler, An introduction to continuity, extrema and related topics for general Gaussian processes. Inst. Math. Statist. Lect. Notes-Monograph Ser. 12 (1990). | MR | Zbl

[2] J.-M. Azais, E. Gassiat C. and Mercadier, Asymptotic distribution and power of the likelihood ratio test for mixtures: bounded and unbounded case. Bernoulli 12 (2006) 775-799. | MR | Zbl

[3] P.J. Bickel, C.A.J. Klaassen, Y. Ritov and J.A. Wellner, Efficient and adaptive estimation for semiparametric models. Johns Hopkins Series in the Mathematical Sciences, Johns Hopkins University Press, Baltimore, MD (1993). | MR | Zbl

[4] A. Chambaz, Testing the order of a model. Ann. Statist. 34 (2006) 1166-1203. | MR | Zbl

[5] A. Chambaz, A. Garivier and E. Gassiat, A mdl approach to hmm with Poisson and Gaussian emissions. Application to order identification. Submitted (2005).

[6] H. Chen and J. Chen, Large sample distribution of the likelihood ratio test for normal mixtures, Statist. Probab. Lett. 2 (2001) 125-133. | MR | Zbl

[7] H. Chen and J. Chen, Test for homogeneity in normal mixtures in the presence of a structural parameter. Statist. Sinica 13 (2003) 355-365. | MR | Zbl

[8] J. Chen and J.D. Kalbfleisch, Modified likelihood ratio test in finite mixture models with a structural parameter. J. Stat. Planning Inf. 129 (2005) 93-107. | MR | Zbl

[9] H. Chen, J. Chen and J.D. Kalbfleisch, A modified likelihood ratio test for homogeneity in finite mixture models. J. Roy. Statist. Soc. B 63 (2001) 19-29. | MR | Zbl

[10] H. Chen, J. Chen and J.D. Kalbfleisch, Testing for a finite mixture model with two components. J. Roy. Statist. Soc. B 66 (2004) 95-115. | MR | Zbl

[11] H. Chernoff and E. Lander, Asymptotic distribution of the likelihood ratio test that a mixture of two binomials is a single binomial. J. Stat. Planning Inf. 43 (1995) 19-40. | MR | Zbl

[12] T. Chihara, An introduction to orthogonal polynomials. Gordon and Breach, New York (1978). | MR | Zbl

[13] G. Ciuperca, Likelihood ratio statistic for exponential mixtures. Ann. Inst. Statist. Math. 54 (2002) 585-594. | MR | Zbl

[14] D. Dacunha-Castelle and E. Gassiat, Testing in locally conic models, and application to mixture models. ESAIM Probab. Statist. 1 (1997) 285-317. | Numdam | MR | Zbl

[15] D. Dacunha-Castelle and E. Gassiat, Testing the order of a model using locally conic parameterization: population mixtures and stationary ARMA processes. Ann. Statist. 27 (1999) 1178-1209. | MR | Zbl

[16] C. Delmas, On likelihood ratio test in Gaussian mixture models, Sankya 65 (2003) 513-531.

[17] B. Garel, Likelihood Ratio Test for Univariate Gaussian Mixture. J. Statist. Planning Inf. 96 (2001) 325-350. | MR | Zbl

[18] B. Garel, Asymptotic theory of the likelihood ratio test for the identification of a mixture. J. Statist. Planning Inf. 131 (2005) 271-296. | MR | Zbl

[19] E. Gassiat, Likelihood ratio inequalities with applications to various mixtures. Ann. Inst. H. Poincaré Probab. Statist. 6 (2002) 897-906. | Numdam | MR | Zbl

[20] E. Gassiat and C. Keribin, The likelihood ratio test for the number of components in a mixture with Markov regime, 2000. ESAIM Probab. Stat. 4 (2000) 25-52. | Numdam | MR | Zbl

[21] J. Ghosh and P. Sen, On the asymptotic performance of the log likelihood ratio statistic for the mixture model and related results, Proceedings of the Berkeley Conference in Honor of Jerzy Neyman and Jack Kiefer, Vol. II. Wadsworth, Belmont, CA (1985) 789-806. | MR

[22] P. Hall and M. Stewart, Theoretical analysis of power in a two-component normal mixture model. J. Statist. Planning Inf. 134 (2005) 158-179. | MR | Zbl

[23] J.A. Hartigan, A failure of likelihood asymptotics for normal mixtures, In Proceedings of the Berkeley conference in honor of Jerzy Neyman and Jack Kiefer (Berkeley, CA, 1983), Vol. II. Wadsworth, Belmont, CA (1985) 807-810. | MR

[24] J. Henna, Estimation of the number of components of finite mixtures of multivariate distributions. Ann. Inst. Statist. Math. 57 (2005) 655-664. | MR | Zbl

[25] L.F. James, C.E. Priebe and D.J. Marchette, Consistent Estimation of Mixture Complexity. Ann. Statist. 29 (2001) 1281-1296. | MR | Zbl

[26] C. Keribin, Consistent estimation of the order of mixture models. Sankhyā Ser. A 62 (2000) 49-66. | MR | Zbl

[27] M. Lemdani and O. Pons, Likelihood ratio test for genetic linkage. Statis. Probab. Lett. 33 (1997) 15-22. | MR | Zbl

[28] M. Lemdani and O. Pons, Likelihood ratio in contamination models. Bernoulli 5 (1999) 705-719. | MR | Zbl

[29] B.G. Lindsay, Mixture models: Theory, geometry, and applications. NSF-CBMS Regional Conf. Ser. Probab. Statist., Vol. 5. Hayward, CA, Institute for Mathematical Statistics (1995). | Zbl

[30] X. Liu and Y. Shao, Asymptotics for the likelihood ratio test in two-component normal mixture models. J. Statist. Planning Inf. 123 (2004) 61-81. | MR | Zbl

[31] X. Liu, C. Pasarica and Y. Shao, Testing homogeneity in gamma mixture models. Scand. J. Statist. 30 (2003) 227-239. | MR | Zbl

[32] Y. Lo, Likelihood ratio tests of the number of components in a normal mixture with unequal variances. Statis. Probab. Lett. 71 (2005) 225-235. | MR | Zbl

[33] F. Lord, Estimating the true-score distributions in psychological testing (an empirical bayes estimation problem). Psychometrika 34 (1969) 259-299. | Zbl

[34] G. Mclachlan and D. Peel, Finite mixture models Wiley Series in Probability and Statistics: Applied Probability and Statistics. Wiley-Interscience, New York (2000). | MR | Zbl

[35] C. Mercadier (2005), toolbox MATLAB. http://www.math.univ-lyon1.fr/mercadier/MAGP/

[36] N. Misra, H. Singh and E.J. Harner, Stochastic comparisons of poisson and binomial random varaibles with their mixtures. Statist. Probab. Lett. 65 279-290. | MR | Zbl

[37] S.A. Murphy and A.W. Van Der Vaart, Semiparametric likelihood ratio inference. Ann. Statist. 25 (1997) 1471-1509. | MR | Zbl

[38] Y.S. Quin and B. Smith, Likelihood ratio test for homogeneity in normal mixtures in the presence of a structural parameter. Statist. Sinica 143 (2004) 1165-1177. | MR | Zbl

[39] Y.S. Quin and B. Smith, The likelihood ratio test for homogeneity in bivariate normal mixtures. J. Multivariate Anal. 97 (2006) 474-491. | MR | Zbl

[40] D.M. Titterington, A.F.M. Smith and U.E. Makov, Statistical analysis of finite mixture distributions. Wiley Series in Probability and Mathematical Statistics: Applied Probability and Statistics. John Wiley & Sons, Ltd (1985). | MR | Zbl

[41] A.W. Van Der Vaart and J.A. Wellner, Weak convergence and empirical processes, Springer Ser. Statist. Springer-Verlag (1996). | MR | Zbl

[42] A.W. Van Der Vaart, Asymptotic statistics, Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, Cambridge (1998). | MR | Zbl

[43] A.W. Van Der Vaart, Semiparametric Statistics, Lectures on probability theory and statistics, Saint-Flour, 1999. Lect. Notes Math. 1781 331-457. Springer, Berlin (2002). | MR | Zbl

[44] G.R. Wood, Binomial mixtures: geometric estimation of the mixing distribution. Ann. Statist. 5 (1999) 1706-1721. | MR | Zbl

Cité par Sources :