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

Abstract

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).Mathematics Subject Classification. 62B10, 62B15, 62M07.

Identifiability of parameters in latent structure models with many observed variables

“…We offer a new proof, based on Kruskal's theorem, of this well-known result. This provides an interesting alternative to Petrie's more direct approach, and one that might extend to more complex frameworks, such as Markov chains with Markov regime, where no identifiability results are known (see, for instance, [9]). Moreover, as a by-product, our approach establishes a new bound on the number of consecutive variables needed, such that the marginal distribution for a generic HMM uniquely determines the full probability distribution.…”

“…This independence in both rows and columns leads to the tensor decomposition of B i . Now since A has full row rank, (9) implies that B i does as well.…”

Hidden Markov model for parameter estimation of a random walk in a Markov environment

“…algorithmic procedures), the statistical properties of these models are slightly more difficult to obtain, (see e.g. Chambaz and Matias, 2009, for model selection issues). As for the convergence properties of the MLE, only the article by Douc, Moulines and Rydén (2004) considers the autoregressive case (instead of HMM) explaining why we focus on their results in our context.…”

Penalized estimate of the number of states in Gaussian linear AR with Markov regime

“…Based on this former work, Gassiat and Boucheron [10] have introduced considerable advances: they proved the strong consistency of the penalized estimator without assuming a priori upper bounds for the number of states; in addition, they showed that the probabilities for underestimating as well as for overestimating fall at an exponential rate with sample size. For AR-MR processes with observations belonging to a finite set the techniques introduced by Gassiat and Boucheron were further used by Chambaz and Matias [4] to simultaneously show the consistency of the number of states of the hidden chain and the memory of the observed process.…”