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 : https://doi.org/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},
     zbl = {1180.62069},
     mrnumber = {2528086},
     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
DA  - 2009///
SP  - 301
EP  - 327
VL  - 13
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/ps:2008010/
UR  - https://zbmath.org/?q=an%3A1180.62069
UR  - https://www.ams.org/mathscinet-getitem?mr=2528086
UR  - https://doi.org/10.1051/ps:2008010
DO  - 10.1051/ps:2008010
LA  - en
ID  - PS_2009__13__301_0
ER  - 
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 1088478 | Zbl 0747.60039

[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 2265342 | Zbl 1134.62010

[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 1245941 | Zbl 0786.62001

[4] A. Chambaz, Testing the order of a model. Ann. Statist. 34 (2006) 1166-1203. | MR 2278355 | Zbl 1096.62016

[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 1841402 | Zbl 0981.62015

[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 1977730 | Zbl 1015.62015

[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 2126840 | Zbl 1058.62020

[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 1811988 | Zbl 0976.62011

[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 2035761 | Zbl 1061.62025

[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 1314126 | Zbl 0812.62015

[12] T. Chihara, An introduction to orthogonal polynomials. Gordon and Breach, New York (1978). | MR 481884 | Zbl 0389.33008

[13] G. Ciuperca, Likelihood ratio statistic for exponential mixtures. Ann. Inst. Statist. Math. 54 (2002) 585-594. | MR 1932403 | Zbl 1013.62018

[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 1468112 | Zbl 1007.62507

[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 1740115 | Zbl 0957.62073

[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 1842105 | Zbl 0972.62011

[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 2139373 | Zbl 1061.62028

[19] E. Gassiat, Likelihood ratio inequalities with applications to various mixtures. Ann. Inst. H. Poincaré Probab. Statist. 6 (2002) 897-906. | Numdam | MR 1955343 | Zbl 1011.62025

[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 1780964 | Zbl 0982.62016

[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 822065

[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 2146091 | Zbl 1066.62031

[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 822066

[24] J. Henna, Estimation of the number of components of finite mixtures of multivariate distributions. Ann. Inst. Statist. Math. 57 (2005) 655-664. | MR 2213484 | Zbl 1094.62067

[25] L.F. James, C.E. Priebe and D.J. Marchette, Consistent Estimation of Mixture Complexity. Ann. Statist. 29 (2001) 1281-1296. | MR 1873331 | Zbl 1043.62023

[26] C. Keribin, Consistent estimation of the order of mixture models. Sankhyā Ser. A 62 (2000) 49-66. | MR 1769735 | Zbl 1081.62516

[27] M. Lemdani and O. Pons, Likelihood ratio test for genetic linkage. Statis. Probab. Lett. 33 (1997) 15-22. | MR 1451126 | Zbl 0902.62128

[28] M. Lemdani and O. Pons, Likelihood ratio in contamination models. Bernoulli 5 (1999) 705-719. | MR 1704563 | Zbl 0929.62015

[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 1163.62326

[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 2058122 | Zbl 1050.62025

[31] X. Liu, C. Pasarica and Y. Shao, Testing homogeneity in gamma mixture models. Scand. J. Statist. 30 (2003) 227-239. | MR 1965104 | Zbl 1034.62010

[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 2126407 | Zbl 1065.62024

[33] F. Lord, Estimating the true-score distributions in psychological testing (an empirical bayes estimation problem). Psychometrika 34 (1969) 259-299. | Zbl 0177.23701

[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 1789474 | Zbl 0963.62061

[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 2039874 | Zbl 1116.60314

[37] S.A. Murphy and A.W. Van Der Vaart, Semiparametric likelihood ratio inference. Ann. Statist. 25 (1997) 1471-1509. | MR 1463562 | Zbl 0928.62036

[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 2126346 | Zbl 1060.62021

[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 2234033 | Zbl 1085.62075

[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 838090 | Zbl 0646.62013

[41] A.W. Van Der Vaart and J.A. Wellner, Weak convergence and empirical processes, Springer Ser. Statist. Springer-Verlag (1996). | MR 1385671 | Zbl 0862.60002

[42] A.W. Van Der Vaart, Asymptotic statistics, Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, Cambridge (1998). | MR 1652247 | Zbl 0910.62001

[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 1915446 | Zbl 1013.62031

[44] G.R. Wood, Binomial mixtures: geometric estimation of the mixing distribution. Ann. Statist. 5 (1999) 1706-1721. | MR 1742506 | Zbl 0955.62033

Cité par Sources :