ON BAHADUR EFFICIENCY OF THE MAXIMUM LIKELIHOOD ESTIMATOR IN HIDDEN MARKOV MODELS
Cheng–Der Fuh · 2004
In this paper, we study large deviations of maximum likelihood and re- lated estimators for hidden Markov models. A hidden Markov model consists of parameterized Markov chains in a Markovian random environment, with the un- derlying environmental Markov chain viewed as missing data. A difficulty with parameter estimation in this model is the non-additivity of the log-likelihood func- tion. Based on a device used to represent the likelihood function as the L1-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 are applied to several types of hidden Markov models commonly used in speech recognition, molecular biology and economics.