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An asymptotic formula for the Bayes risk in discriminating between two Markov chains

Part of: Limit theorems Inference from stochastic processes

Published online by Cambridge University Press:  14 July 2016

A. V. Nagaev
A. V. Nagaev*
Affiliation:
Nicolaus Copernicus University, Toruń
*
1Postal address: Faculty of Mathematics and Informatics, Nicolaus Copernicus University, Toruń, Poland. Email: nagaev@mat.uni.torun.pl
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Abstract

The problem of discriminating between two Markov chains is considered. It is assumed that the common state space of the chains is finite and all the finite dimensional distributions are mutually absolutely continuous. The Bayes risk is expressed through large deviation probabilities for sums of random variables defined on an auxiliary Markov chain. The proofs are based on a large deviation theorem recently established by Z. Szewczak.

Keywords

Cramér large deviationsPerron–Frobenius theoremlikelihood ratioprimitive matrixspectral radius

MSC classification

Primary: 62M02: Markov processes
Secondary: 60F10: Large deviations
Type
Estimation problems
Information
Journal of Applied Probability , Volume 38 , Issue A: Probability, Statistics and Seismology , 2001 , pp. 131 - 141
Copyright
Copyright © Applied Probability Trust 2001 

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References

Borovkov, A. A. and Mogulski, A. A. (1992). Large deviations and testing statistical hypotheses, II: Large deviations of maximum points of random fields. Siberian Adv. Math. 2, 43–72.Google Scholar
Borovkov, A. A. and Mogulski, A. A. (1993). Large deviations and testing statistical hypotheses, III: Asymptotically optimal tests for composite hypotheses. Siberian Adv. Math. 3, 19–86.Google Scholar
Chung, K. L. (1967). Markov Chains with Stationary Transition Probabilities. Springer, Heidelberg.Google Scholar
Dembo, A. and Zeitouni, O. (1993). Large Deviation Techniques and Applications. Jones and Bartlett, Boston, MA.Google Scholar
Doob, J. L. (1953). Stochastic Processes. Wiley, New York.Google Scholar
Horn, R. A. and Johnson, C. R. (1986). Matrix Analysis. Cambridge University Press.Google Scholar
Nagaev, A. V. (1996). Limit Theorems under Testing Hypotheses. Uniwersytet Mikolaja Kopernika, Torun.Google Scholar
Nagaev, S. V. (1961). More exact statements of limit theorems for homogeneous Markov chains. Teor. Veroyatnost. i Primenen. 6, 67–86 (in Russian). English translation: Theory Probab. Appl. 6, 62-81.Google Scholar
Scheffel, P. and Von Weizsacker, H. (1997). On risk rates and large deviations in finite Markov chain experiments. Math. Methods Statist. 6, 293–312.Google Scholar
Szewczak, Z. (2000). Remark on a large deviation theorem for Markov chain with a finite number of states. Preprint, Faculty of Mathematics and Informatics, Uniwersytet Mikolaja Kopernika, Torun.Google Scholar