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TESTING FOR REVERSIBILITY IN MARKOV CHAIN DATA

Published online by Cambridge University Press:  30 July 2012

Tara L. Steuber ,
Peter C. Kiessler  and
Robert Lund
Tara L. Steuber
Affiliation:
Department of Mathematical Sciences, Clemson University, O-106 Martin Hall, Clemson, SC 29634-0975 E-mails: tsteube@clemson.edu; kiesslp@clemson.edu; lund@clemson.edu
Peter C. Kiessler
Affiliation:
Department of Mathematical Sciences, Clemson University, O-106 Martin Hall, Clemson, SC 29634-0975 E-mails: tsteube@clemson.edu; kiesslp@clemson.edu; lund@clemson.edu
Robert Lund
Affiliation:
Department of Mathematical Sciences, Clemson University, O-106 Martin Hall, Clemson, SC 29634-0975 E-mails: tsteube@clemson.edu; kiesslp@clemson.edu; lund@clemson.edu
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Abstract

This paper introduces two statistics that assess whether (or not) a sequence sampled from a stationary time-homogeneous Markov chain on a finite state space is reversible. The test statistics are based on observed deviations of transition sample counts between each pair of states in the chain. First, the joint asymptotic normality of these sample counts is established. This result is then used to construct two chi-squared-based tests for reversibility. Simulations assess the power and type one error of the proposed tests.

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 26 , Issue 4 , October 2012 , pp. 593 - 611
Copyright
Copyright © Cambridge University Press 2012

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References

1.Anderson, T.W. & Goodman, L. A. (1957). Statistical inference about Markov chains. The Annals of Mathematical Statistics 28: 89–110.CrossRefGoogle Scholar
2.Annis, D. H., Kiessler, P. C., Lund, R. & Steuber, T. L. (2010). Estimation in reversible Markov chains. The American Statistician 64: 116–120.CrossRefGoogle Scholar
3.Basawa, I. V. & Rao, B. L. S. (1980). Statistical inference for stochastic processes. New York: Academic Press, Inc.Google Scholar
4.Billingsley, P. (1961). Statistical methods in Markov chains. The Annals of Mathematical Statistics 32: 12–40.CrossRefGoogle Scholar
5.Derman, C. (1956). Some asymptotic distribution theory for Markov chains with a denumerable number of states. Biometrika 43: 285–294.CrossRefGoogle Scholar
6.Diaconis, P. & Rolles, S. W. W. (2006). Bayesian analysis for reversible Markov chains. The Annals of Statistics 34: 1270–1292.CrossRefGoogle Scholar
7.Di Cecco, D. (2011). Conditional exact tests for Markovianity and reversibility in multiple categorical sequences. TEST 21: 170–187.CrossRefGoogle Scholar
8.Edwards, D. G. (1980). Large sample tests for stationarity and reversibility in finite Markov chains. Scandinavian Journal of Statistics 7: 203–206.Google Scholar
9.Graybill, F. A. (1976). Theory and application of the linear model. Pacific Grove, CA: Duxbury.Google Scholar
10.Greenwood, P.E. & Wefelmeyer, W. (1999). Reversible Markov chains and optimality of symmetrized empirical estimators. Bernoulli, 5, 109–123.CrossRefGoogle Scholar
11.Kijima, M. (1997). Markov processes for stochastic modeling. London: Chapman & Hall.CrossRefGoogle Scholar
12.Richman, D. & Sharp, W.E. (1990). A method for determining the reversibility of a Markov sequence. Mathematical Geology 22: 749–761.CrossRefGoogle Scholar
13.Ross, S. M. (2007). Introduction to Probability Models, 9th ed.Burlington, MA: Academic Press.Google Scholar
14.Sharifdoost, M., Mahmoodi, S. & Pasha, E. (2009). A statistical test for time reversibility of stationary finite state Markov chains. Applied Mathematical Sciences 3: 2563–2574.Google Scholar
15.Stroock, D. (2005). An introduction to Markov processes. New York: Springer.Google Scholar
16.Ash, R.B. & Doleans-Dade, C.A. (2000). Probability and measure theory. New York: Academic Press Inc.Google Scholar