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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
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[1] Albert, A., Estimating the infinitesimal generator of a continuous time finite state Markov process, Ann. math. statist., 33, 727-753, (1962) · Zbl 0119.14702
[2] Anderson, T.; Goodman, L., Statistical inference about Markov chains, Ann. math. statist., 28, 89-110, (1957) · Zbl 0087.14905
[3] Basharin, G., On a statistical estimation for the entropy of a sequence of independent random variables, Theor. probab. appl., 4, 333-336, (1959)
[4] Bladt, M.; Sørensen, M., Statistical inference for discretely observed Markov jump processes, J. roy. statist. soc. B, 67, 395-410, (2005) · Zbl 1069.62061
[5] Chiquet, J.; Limnios, N., Estimating stochastic dynamical systems driven by a continuous-time jump Markov process, Methodol. comput. appl. probab., 8, 431-447, (2006) · Zbl 1118.62087
[6] Ciuperca, G.; Girardin, V., Estimation of the entropy rate of a countable Markov chain, Comm. statist. theory methods, 36, 1-15, (2007)
[7] Cover, T.; Thomas, J., Elements of information theory, (2006), Wiley New Jersey
[8] Crommelin, D.; Vanden-Eijnden, E., Data-based inference of generators for Markov jump processes using convex optimization, Multiscale model. simul., 7, 1751-1778, (2009) · Zbl 1182.62163
[9] El-Affendi, M.; Kouvastos, D., A maximum entropy analysis of the M/G/1 and G/M/1 queueing systems at equilibrium, Acta inform., 19, 339-355, (1983) · Zbl 0494.60095
[10] Gao, Y.; Kontoyiannis, I.; Bienenstosk, E., Estimating the entropy of binary time series: methodology, some theory and a simulation study, Entropy, 10, 71-99, (2008) · Zbl 1179.94045
[11] Girardin, V.; Sesboüé, A., Comparative construction of plug-in estimators of the entropy rate of two-state Markov chains, Methodol. comput. appl. probab., 11, 2, 181-200, (2009) · Zbl 1163.62342
[12] Guiasu, S., Maximum entropy condition in queueing theory, J. oper. res. soc., 37, 3, 293-301, (1986) · Zbl 0582.60090
[13] Harris, B., The statistical estimation of entropy in the non-parametric case, Colloq. math. soc. janos bolyai, 16, 323-355, (1977) · Zbl 0364.94025
[14] Jaynes, E., Information theory and statistical mechanics, Phys. rev., 106, 620-630, (1957) · Zbl 0084.43701
[15] Kingman, J., The imbedding problem for finite Markov chains, Z. wahrscheinlichkeitstheor., 1, 14-24, (1962) · Zbl 0118.13501
[16] Metzner, P.; Horenko, I.; Schütte, C., Generator estimation of Markov jump processes based on incomplete observations non-equidistant in time, Phys. rev. E, 76, 066702, (2007)
[17] Regnault, P., 2009. Plug-in estimator of the entropy rate of a pure-jump two-state Markov process. In: Bayesian Inference and Maximum Entropy Methods in Science and Engineering. AIP Conference Proceedings, vol. 1193, pp. 153-160.
[18] Shannon, C., A mathematical theory of communication, The Bell system technical journal, 27, 379-423, (1948), 623-656 · Zbl 1154.94303
[19] Shao, J., 2003. Mathematical Statistics, second ed. Springer, New-York. · Zbl 1018.62001
[20] Taga, Y., On the limiting distributions in Markov renewal processes with finitely many states, Ann. inst. statist. math., 15, 1-10, (1963) · Zbl 0168.16203
[21] Zubkov, A., Limit distribution for a statistical estimator of the entropy, Theor. probab. appl., 18, 611-618, (1973) · Zbl 0294.62031
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