The Shannon marginal entroy and entropy rate of ergodic Markov chains are explicit functions of the coefficients of their transition matrices. Plug-in estimators of entropy can thus be constructed from estimators of these coefficients.
For generalized entropies such as Rényi or Tsallis, a closed form expression of the entropy rate - when meaningful, can be obtained and estimated by means of operator theory methods.
Different schemes of observation are considered, for continuous and discrete time processes with binary, finite or denumerable state spaces. All the constructed estimators behave asymptotically well, with limit distributions depending on explicit parameters.
Estimation of the Entropy Rate of a Countable Markov Chain, Communication in Statistics: Theory and Methods, V36, pp2543-2557, G. Ciuperca and V. Girardin (2007)
Comparative Construction of Plug-in Estimators of the Entropy Rate of Two-State Markov Chains, Methodology and Computing in Applied Probability, V11, pp. 181-200, V. Girardin and A. Sesboü (2009)
Computation of Generalized Entropy Rates. Application and Estimation for Countable Markov Chains, IEEE Transactions on Information Theory, to appear, G. Ciuperca, V. Girardin et L. Lhote (2011)