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International audienceThis 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 of states of the hidden Markov chain and the memory of the conditional Markov chain. We study the properties of penalized maximum likelihood estimators for the unknown order 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 and . 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 (that is for hidden Markov models)

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