Consistency of the maximum likelihood estimator for general hidden Markov models

Abstract

Consider 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 chains.

Published in at http://dx.doi.org/10.1214/10-AOS834 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (4)

• Asymptotic properties of the maximum likelihood estimator in autoregressive models with Markov regime
• The stability of conditional Markov processes and Markov chains in random environments
• Efficient likelihood estimation in state space models
• On some problems in the article "Efficient Likelihood Estimation in State Space Models" by Cheng-Der Fuh

Cited by in corpus (5)

• Asymptotic properties of the maximum likelihood estimation in misspecified hidden Markov models
• The Bernstein-von Mises theorem and nonregular models
• A guide to state-space modeling of ecological time series
• Consistency of maximum likelihood estimation for some dynamical systems
• Asymptotic properties of the maximum likelihood estimator for Hidden Markov Models indexed by binary trees