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Number of hidden states and memory: a joint order estimation problem for Markov chains with Markov regime

Published online by Cambridge University Press:  21 February 2009

Antoine Chambaz  and
Catherine Matias
Antoine Chambaz
Affiliation:
Laboratoire MAP5, UMR CNRS 8145, Université René Descartes, 45 rue des Saints-Pères, 75270 Paris Cedex 06, France; [email protected]
Catherine Matias
Affiliation:
Laboratoire Statistique et Génome, UMR CNRS 8071, Tour Évry 2, 523 pl. des Terrasses de l'Agora, 91000 Évry, France; [email protected]
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Abstract

This paper deals with order identification for Markov chains with Markov regime (MCMR) in the context of finite alphabets. We define the joint order of a MCMR process in terms of the number k of states of the hidden Markov chain and the memory m of the conditional Markov chain. We study the properties of penalized maximum likelihood estimators for the unknown order (k,m) of an observed MCMR process, relying on information theoretic arguments. The novelty of our work relies in the joint estimation of two structural parameters. Furthermore, the different models in competition are not nested. In an asymptotic framework, we prove that a penalized maximum likelihood estimator is strongly consistent without prior bounds on k and m. We complement our theoretical work with a simulation study of its behaviour. We also study numerically the behaviour of the BIC criterion. A theoretical proof of its consistency seems to us presently out of reach for MCMR, as such a result does not yet exist in the simpler case where m = 0 (that is for hidden Markov models).

Keywords

Markov regimeorder estimationhidden statesconditional memoryhidden Markov model
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 13 , July 2009 , pp. 38 - 50
Copyright
© EDP Sciences, SMAI, 2009

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