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The likelihood ratio test for the number of componentsin amixture with Markov regime

Published online by Cambridge University Press:  15 August 2002

Elisabeth Gassiat  and
Christine Keribin
Elisabeth Gassiat
Affiliation:
Laboratoire Modélisation Stochastique et Statistique, Université d'Orsay, bâtiment 425, 91405 Orsay, France; [email protected].
Christine Keribin
Affiliation:
Laboratoire Modélisation Stochastique et Statistique, Université d'Orsay, bâtiment 425, 91405 Orsay, France; [email protected].
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Abstract

We study the LRT statistic for testing a single population i.i.d. model against a mixture of two populations with Markov regime.We prove that the LRT statistic converges to infinity in probability as the number of observations tends to infinity. This is a consequence of a convergence result of the LRT statistic for a subproblem where the parameters are restricted to a subset of the whole parameter set.

Keywords

Likelihood ratio testhidden Markov modelorder of a mixture.
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 4 , 2000 , pp. 25 - 52
Copyright
© EDP Sciences, SMAI, 2000

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References

Atwood, L.D., Wilson, A.F., Bailey-Wilson, J.E., Carruth, J.N. and Elston, R.C., On the distribution of the likelihood ratio test statistic for a mixture of two normal distributions. Comm. Statist. Simulation Comput. 25 (1996) 733-740. CrossRef
Baum, L.E. and Petrie, T., Statistical inference for probabilistic functions of finite state Markov chains. Ann. Math. Stat. 37 (1966) 1554-1563. CrossRef
Bickel, P.J. and Ritov, Y., Inference in hidden Markov models I: Local asymptotic normality in the stationary case. Bernoulli 2 (1996) 199-228. CrossRef
Bickel, P.J., Ritov, Y. and Asymptotic, T. Ryden normality of the maximum-likelihood estimator for general hidden Markov models. Annals of Stat. 26 (1998) 1614-1635.
Chuang, R.-J. and Mendell, N.R., The approximate null distribution of the likelihood ratio test for a mixture of two bivariate normal distributions with equal covariance. Comm. Statist. Simulation Comput. 26 (1997) 631-648. CrossRef
Churchill, G.A., Stochastic models for heterogeneous DNA sequences. Bull. Math. Biology 51 (1989) 79-94. CrossRef
G. Ciuperca, Sur le test de maximum de vraisemblance pour le mélange de populations. Note aux C.R.A.S., 328, Série I, 4 (1999) 351-358.
D. Dacunha-Castelle and M. Duflo, Probabilités et statistiques, Tome 2. Masson (1993).
Dacunha-Castelle, D. and Gassiat, E., Estimation of the number of components in a mixture. Bernoulli 3 (1997a) 279-299. CrossRef
D. Dacunha-Castelle and E. Gassiat, Testing in locally conic models. ESAIM Probab. Statist. 1 (1997b).
Dacunha-Castelle, D. and Gassiat, E., Testing the order of a model using locally conic parametrization: Population mixtures and stationary ARMA processes. Ann. Statist. 27 (1999) 1178-1209.
Dempster, A.P., Laird, N.M. and Rubin, D.B., Large Maximum-likelihood from incomplete data via the EM algorithm. J. Roy. Statist. Soc. Ser. B 39 (1977) 1-38.
R. Douc and C. Matias, Asymptotics of the Maximum Likelihood Estimator for general Hidden Markov Models (1999) (submitted).
M. Duflo, Algorithmes stochastiques. Springer (1996).
Feng, Z.D. and McCulloch, C.E., Using bootstrap Likelihood Ratio in Finite Mixture Models. J. Roy. Statist. Soc. Ser. B 58 (1996) 609-617.
L. Finesso, Consistent Estimation of the Order for Markov and Hidden Markov Chains. Ph.D. Thesis, University of Maryland (1990).
Fredkin, D.R. and Rice, J.A., Maximum likelihood estimation and identification directly from single-channel recordings. Proc. Roy. Soc. London Ser. B 249 (1992) 125-132. CrossRef
P. Hall and C.C. Heyde, Martingale Limit Theory and Its Application. Academic Press (1980).
J.A. Hartigan, A failure of likelihood ratio asymptotics for normal mixtures, in Proc. Berkeley Conference in Honor of Jerzy Neyman and Jack Kiefer, edited by L.M. Le Cam and R.A. Olshen (1985) 807-810.
Henna, J., On estimating the number of constituents of a finite mixture of continuous distributions. Ann. Inst. Statist. Math. 37 (1985) 235-240. CrossRef
Jensen, J.L. and Petersen, N.V., Asymptotic normality of the maximum likelihood estimator in state space models. Ann. Statist. 27 (1999) 514-535.
C. Keribin, Tests de modèles par maximum de vraisemblance, Thèse de l'Université d'Evry-Val d'Essonne (1999).
C. Keribin, Consistent estimation of the Order of Mixture Models (1997) (submitted).
Leroux, B.G., Maximum-likelihood estimation for hidden Markov models. Stochastic Process Appl. 40 (1992) 127-143. CrossRef
Leroux, B.G. and Puterman, M.L., Maximum Penalized Likelihood Estimation for Independent and Markov-Dependent Mixture Models. Biometrics 48 (1992) 545-558. CrossRef
B.G. Lindsay, Mixture models: Theory, Geometry and Applications (1995).
I.L. Mac Donald and W. Zucchini, Hidden Markov and Other Models for Discrete-valued Time Series. Chapman and Hall (1997).
McLachlan, G.J., Bootstrapping, On the Likelihood Ratio Test Statistic for the Number of Components in a Normal Mixture. Appl. Statist. 36 (1987) 318-324. CrossRef
L. Mevel, Statistique asymptotique pour les modèles de Markov cachés. Thèse de l'Université de Rennes I (1997).
L. Mevel and F. LeGland, Exponential forgetting and Geometric Ergodicity in Hidden Markov models. Math. Control Signals Systems (to appear).
S.P. Meyn and R.L. Tweedie, Markov chains and stochastic stability. Springer-Verlag (1993).
Rabiner, L.R., A tutorial on hidden Markov models and selected applications in speech recognition. Proc. IEEE 77 (1989) 257-284. CrossRef
Estimating, T. Ryden the order of hidden Markov models. Statistics 26 (1995) 345-354.
P. Vandekerkhove, Identification de l'ordre des processus ARMA stables. Contribution à l'étude statistique des chaînes de Markov cachées. Thèse de l'Université de Montpellier II (1997).
A. Van der Vaart, Asymptotic Statistics. Cambridge Ed. (1999).