On Bahadur efficiency of the maximum likelihood estimator in hidden Markov models

In this paper, we study large deviations of maximum likelihood and related estimators for hidden Markov models. A hidden Markov model consists of parameterized Markov chains in a Markovian random environment, with the underlying environmental Markov chain viewed as missing data. A difficulty with parameter estimation in this model is the non-additivity of the log-likelihood function. Based on a device used to represent the likelihood function as the L-1-norm of products of Markov random matrices, we investigate the tail probabilities for consistent estimators in hidden Markov models. The main result is that, under some regularity conditions, the maximum likelihood estimator is an asymptotically locally optimal estimator in Bahadur's sense. The results axe applied to several types of hidden Markov models commonly used in speech recognition, molecular biology and economics.