Hostname: page-component-84b7d79bbc-2l2gl Total loading time: 0 Render date: 2024-07-25T10:29:18.629Z Has data issue: false hasContentIssue false
  • English
  • Français

The likelihood ratio test for general mixture models withor without structural parameter

Published online by Cambridge University Press:  21 July 2009

Jean-Marc Azaïs ,
Élisabeth Gassiat  and
Cécile Mercadier
Jean-Marc Azaïs
Affiliation:
Institut de Mathématiques de Toulouse, UMR 5219, Université Paul Sabatier, 31062 Toulouse Cedex 9, France; azais@cict.fr
Élisabeth Gassiat
Affiliation:
Équipe Probabilités, Statistique et Modélisation, UMR CNRS 8628, Université Paris-Sud, Bâtiment 425, Université de Paris-Sud, 91405 Orsay Cedex, France.
Cécile Mercadier
Affiliation:
Université de Lyon, Université Lyon 1, CNRS UMR 5208 Institut Camille Jordan, Bâtiment du Doyen Jean Braconnier, 43, bd du 11 Novembre 1918, 69622 Villeurbanne Cedex, France.
Get access

Abstract

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.

Keywords

Likelihood ratio testmixture modelsnumber of componentslocal powercontiguity
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 13 , July 2009 , pp. 301 - 327
Copyright
© EDP Sciences, SMAI, 2009

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

R.J. Adler, An introduction to continuity, extrema and related topics for general Gaussian processes. Inst. Math. Statist. Lect. Notes-Monograph Ser. 12 (1990).
Azais, J.-M., Gassiat, E. Mercadier, C. , Asymptotic distribution and power of the likelihood ratio test for mixtures: bounded and unbounded case. Bernoulli 12 (2006) 775–799. CrossRef
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).
A. Chambaz, Testing the order of a model. Ann. Statist. 34 (2006) 1166–1203.
A. Chambaz, A. Garivier and E. Gassiat, A mdl approach to hmm with Poisson and Gaussian emissions. Application to order identification. Submitted (2005).
H. Chen and J. Chen, Large sample distribution of the likelihood ratio test for normal mixtures, Statist. Probab. Lett. 2 (2001) 125–133.
Chen, H. and Chen, J., Test for homogeneity in normal mixtures in the presence of a structural parameter. Statist. Sinica 13 (2003) 355–365.
Chen, J. and Kalbfleisch, J.D., Modified likelihood ratio test in finite mixture models with a structural parameter. J. Stat. Planning Inf. 129 (2005) 93–107. CrossRef
Chen, H., Chen, J. and Kalbfleisch, J.D., A modified likelihood ratio test for homogeneity in finite mixture models. J. Roy. Statist. Soc. B 63 (2001) 19–29. CrossRef
Chen, H., Chen, J. and Kalbfleisch, J.D., Testing for a finite mixture model with two components. J. Roy. Statist. Soc. B 66 (2004) 95–115. CrossRef
Chernoff, H. and Lander, E., 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. CrossRef
T. Chihara, An introduction to orthogonal polynomials. Gordon and Breach, New York (1978).
Ciuperca, G., Likelihood ratio statistic for exponential mixtures. Ann. Inst. Statist. Math. 54 (2002) 585–594. CrossRef
Dacunha-Castelle, D. and Gassiat, E., Testing in locally conic models, and application to mixture models. ESAIM Probab. Statist. 1 (1997) 285–317. CrossRef
Dacunha-Castelle, D. and Gassiat, E., Testing the order of a model using locally conic parameterization: population mixtures and stationary ARMA processes. Ann. Statist. 27 (1999) 1178–1209.
C. Delmas, On likelihood ratio test in Gaussian mixture models, Sankya 65 (2003) 513-531.
Garel, B., Likelihood Ratio Test for Univariate Gaussian Mixture. J. Statist. Planning Inf. 96 (2001) 325–350. CrossRef
Garel, B., Asymptotic theory of the likelihood ratio test for the identification of a mixture. J. Statist. Planning Inf. 131 (2005) 271–296. CrossRef
Gassiat, E., Likelihood ratio inequalities with applications to various mixtures. Ann. Inst. H. Poincaré Probab. Statist. 6 (2002) 897–906. CrossRef
Gassiat, E. and Keribin, C., The likelihood ratio test for the number of components in a mixture with Markov regime, 2000. ESAIM Probab. Stat. 4 (2000) 25–52. CrossRef
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.
Hall, P. and Stewart, M., Theoretical analysis of power in a two-component normal mixture model. J. Statist. Planning Inf. 134 (2005) 158–179. CrossRef
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.
Henna, J., Estimation of the number of components of finite mixtures of multivariate distributions. Ann. Inst. Statist. Math. 57 (2005) 655–664. CrossRef
James, L.F., Priebe, C.E. and Marchette, D.J., Consistent Estimation of Mixture Complexity. Ann. Statist. 29 (2001) 1281–1296.
Keribin, C., Consistent estimation of the order of mixture models. Sankhyā Ser. A 62 (2000) 49–66.
Lemdani, M. and Pons, O., Likelihood ratio test for genetic linkage. Statis. Probab. Lett. 33 (1997) 15–22. CrossRef
Lemdani, M. and Pons, O., Likelihood ratio in contamination models. Bernoulli 5 (1999) 705–719. CrossRef
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).
Liu, X. and Shao, Y., Asymptotics for the likelihood ratio test in two-component normal mixture models. J. Statist. Planning Inf. 123 (2004) 61–81. CrossRef
Liu, X., Pasarica, C. and Shao, Y., Testing homogeneity in gamma mixture models. Scand. J. Statist. 30 (2003) 227–239. CrossRef
Likelihood, Y. Lo ratio tests of the number of components in a normal mixture with unequal variances. Statis. Probab. Lett. 71 (2005) 225–235.
Lord, F., Estimating the true-score distributions in psychological testing (an empirical bayes estimation problem). Psychometrika 34 (1969) 259–299. CrossRef
G. McLachlan and D. Peel, Finite mixture models Wiley Series in Probability and Statistics: Applied Probability and Statistics. Wiley-Interscience, New York (2000).
C. Mercadier (2005), toolbox MATLAB. http://www.math.univ-lyon1.fr/mercadier/MAGP/
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.
Murphy, S.A. and van der Vaart, A.W., Semiparametric likelihood ratio inference. Ann. Statist. 25 (1997) 1471–1509.
Quin, Y.S. and Smith, B., Likelihood ratio test for homogeneity in normal mixtures in the presence of a structural parameter. Statist. Sinica 143 (2004) 1165–1177.
Quin, Y.S. and Smith, B., The likelihood ratio test for homogeneity in bivariate normal mixtures. J. Multivariate Anal. 97 (2006) 474–491.
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).
A.W. van der Vaart and J.A. Wellner, Weak convergence and empirical processes, Springer Ser. Statist. Springer-Verlag (1996).
A.W. van der Vaart, Asymptotic statistics, Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, Cambridge (1998).
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).
G.R. Wood, Binomial mixtures: geometric estimation of the mixing distribution. Ann. Statist. 5 (1999) 1706–1721.