Convergence rates of the maximum likelihood estimator of hidden Markov models

Abstract

We derive the almost sure rate of convergence of the maximum likelihood estimator of the parameters of a hidden Markov model with continuous observations and finite state space. The analysis is based on the geometric ergodicity properties of the prediction filter and its derivatives. As an example of application of these results we prove that, also in this context, the likelihood ratio is a consistent statistic for model selection.