Consistency of the maximum likelihood estimator for general hidden Markov models
Open Access
- 1 February 2011
- journal article
- Published by Institute of Mathematical Statistics in The Annals of Statistics
- Vol. 39 (1)
- https://doi.org/10.1214/10-aos834
Abstract
International audienceConsider a parametrized family of general hidden Markov models, where both the observed and unobserved components take values in a complete separable metric space. We prove that the maximum likelihood estimator (MLE) of the parameter is strongly consistent under a rather minimal set of assumptions. As special cases of our main result, we obtain consistency in a large class of nonlinear state space models, as well as general results on linear Gaussian state space models and finite state models. A novel aspect of our approach is an information-theoretic technique for proving identifiability, which does not require an explicit representation for the relative entropy rate. Our method of proof could therefore form a foundation for the investigation of MLE consistency in more general dependent and non-Markovian time series. Also of independent interest is a general concentration inequality for V-uniformly ergodic Markov chainsKeywords
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