Estimation of the Entropy Rate of a Countable Markov Chain

We consider here ergodic homogeneous Markov chains with countable state spaces. The entropy rate of the chain is an explicit function of its transition and stationary distributions. We construct estimators for this entropy rate and for the entropy of the stationary distribution of the chain, in the parametric and nonparametric cases. We study estimation from one sample with long length and from many independent samples with given length. In the parametric case, the estimators are deduced by plug-in from the maximum likelihood estimator of the parameter. In the nonparametric case, the estimators are deduced by plug-in from the empirical estimators of the transition and stationary distributions. They are proven to have good asymptotic properties.