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Estimation using plug-in of the stationary distribution and Shannon entropy of continuous time Markov processes. (English) Zbl 1213.62009
Summary: A natural way to deal with the uncertainty of an ergodic finite state space Markov process is to investigate the entropy of its stationary distribution. When the process is observed, it becomes necessary to estimate this entropy. We estimate both the stationary distribution and its entropy by plug-in of the estimators of the infinitesimal generator. Three situations of observation are discussed: one long trajectory is observed, several independent short trajectories are observed, or the process is observed at discrete times. The good asymptotic behavior of the plug-in estimators is established. We also illustrate the behavior of the estimators through simulations.

##### MSC:
 62B10 Statistical aspects of information-theoretic topics 62M05 Markov processes: estimation; hidden Markov models 65C60 Computational problems in statistics (MSC2010) 60J27 Continuous-time Markov processes on discrete state spaces
##### Keywords:
ergodicity; plug-in estimation; pure jump Markov processes
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