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On exact and large deviation approximation for the distribution of the longest run in a sequence of two-state Markov dependent trials

Part of: Distribution theory - Probability Markov processes

Published online by Cambridge University Press:  14 July 2016

James C. Fu ,
Liqun Wang  and
W. Y. Wendy Lou
James C. Fu*
Affiliation:
University of Manitoba
Liqun Wang*
Affiliation:
University of Manitoba
W. Y. Wendy Lou*
Affiliation:
University of Toronto
*
∗ Postal address: Department of Statistics, University of Manitoba, Winnipeg, Manitoba R3T 2N2, Canada.
∗ Postal address: Department of Statistics, University of Manitoba, Winnipeg, Manitoba R3T 2N2, Canada.
∗∗∗ Postal address: Department of Public Health Sciences, University of Toronto, Toronto, Ontario M5S 1A8, Canada.
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Abstract

Consider a sequence of outcomes from Markov dependent two-state (success-failure) trials. In this paper, the exact distributions are derived for three longest-run statistics: the longest failure run, longest success run, and the maximum of the two. The method of finite Markov chain imbedding is used to obtain these exact distributions, and their bounds and large deviation approximation are also studied. Numerical comparisons among the exact distributions, bounds, and approximations are provided to illustrate the theoretical results. With some modifications, we show that the results can be easily extended to Markov dependent multistate trials.

Keywords

Success runfailure runMarkov chain imbeddingtransition probability matrixprobability of large deviation

MSC classification

Primary: 60E05: Distributions
Secondary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Type
Research Papers
Information
Journal of Applied Probability , Volume 40 , Issue 2 , June 2003 , pp. 346 - 360
Copyright
Copyright © Applied Probability Trust 2003 

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References

Balakrishnan, N., and Koutras, M. V. (2002). Runs and Scans with Applications. John Wiley, New York.Google Scholar
Barbour, A. D., Chryssaphinou, O., and Roos, M. (1995). Compound Poisson approximation in reliability theory. IEEE Trans. Reliab. 44, 398–402.Google Scholar
Barbour, A. D., Holst, L., and Janson, S. (1992). Poisson Approximations. Oxford University Press.Google Scholar
Boutsikas, M. V., and Koutras, M. V. (2000). Reliability approximation for Markov chain imbeddable systems. Methodology Comput. Appl. Prob. 2, 393–411.Google Scholar
Burr, E. J., and Cane, G. (1961). Longest run of consecutive observations having a special attribute. Biometrika 48, 461–465.Google Scholar
Chao, M. T., and Fu, J. C. (1989). A limit theorem of certain repairable systems. Ann. Inst. Statist. Math. 41, 809–818.Google Scholar
Chao, M. T., and Fu, J. C. (1991). The reliability of a large series system under the Markov structure. Adv. Appl. Prob. 23, 894–908.Google Scholar
Chryssaphinou, O., and Papastavridis, S. G. (1990). Limit distribution for a consecutive-k-out-of-n: F system. Adv. Appl. Prob. 22, 491–493.Google Scholar
Derman, C., Liberman, G. J., and Ross, S. (1982). On the consecutive-k-out-of-n:F systems. IEEE Trans. Reliab. 31, 57–63.Google Scholar
Doi, M., and Yamamoto, E. (1998). On the joint distribution of runs in a sequence of multi-state trials. Statist. Prob. Lett. 39, 133–141.CrossRefGoogle Scholar
Erd Hos, P. and Révész, P. (1975). On the length of the longest head run. In Topics in Information Theory (Colloq. Math. Soc. János Bolyai 16), eds Csiszár, I. and Elias, P., North-Holland, Amsterdam, pp. 219–228.Google Scholar
Fu, J. C. (1986). Bounds for reliability of large consecutive-k-out-of-n: F systems with unequal component probabilities. IEEE Trans. Reliab. 35, 316–319.Google Scholar
Fu, J. C. (1996). Distribution theory of runs and patterns associated with a sequence of multi-state trials. Statist. Sinica 6, 957–974.Google Scholar
Fu, J. C., and Chang, Y. M. (2002). On probability generating functions for waiting time distributions of compound patterns in a sequence of multistate trials. J. Appl. Prob. 39, 70–80.Google Scholar
Fu, J. C., and Koutras, M. V. (1994). Distribution theory of runs: a Markov chain approach. J. Amer. Statist. Soc. 89, 1050–1058.Google Scholar
Gibbons, J. D. (1971). Nonparametric Statistical Inference. McGraw-Hill, New York.Google Scholar
Godbole, A. P., and Schaffner, A. A. (1993). Improved Poisson approximations for word patterns. Adv. Appl. Prob. 25, 334–347.Google Scholar
Goncharov, V. L. (1944). On the field of combinatory analysis. Izv. Akad. Nauk. SSSR Ser. Mat. 8, 3–48 (in Russian). English translation: Amer. Math. Soc. Transl. 19 (1962), 1–46.Google Scholar
Gordon, L., Schilling, M. F., and Waterman, M. S. (1986). An extreme value theory for long head runs. Prob. Theory Relat. Fields 72, 279–287.Google Scholar
Hirano, K. (1986). Some properties of the distributions of order k. In Fibonacci Numbers and Their Applications (Patras, 1984), Reidel, Dordrecht, pp. 43–53.Google Scholar
Koutras, M. V. (1997). Waiting time distributions associated with runs of fixed length in two-state Markov chains. Ann. Inst. Statist. Math. 49, 123–139.Google Scholar
Koutras, M. V., and Alexandrou, V. A. (1997). Sooner waiting time problems in a sequence of trinary trials. J. Appl. Prob. 34, 593–609.Google Scholar
Lou, W. Y. W. (1996). On runs and longest run tests: a method of finite Markov chain imbedding. J. Amer. Statist. Soc. 91, 1595–1601.Google Scholar
Muselli, M. (2000). New improved bounds for reliability of consecutive-k-out-of-n:F systems. J. Appl. Prob. 37, 1164–1170.Google Scholar
Papastavridis, S. G., and Koutras, M. V. (1993). Bounds for reliability of consecutive-k-within-m-out-of-n systems. IEEE Trans. Reliab. 42, 156–160.Google Scholar
Petrov, V. V. (1965). On the probabilities of large deviations for sums of independent random variables. Theory Prob. Appl. 10, 287–298.Google Scholar
Philippou, A. N., and Makri, F. S. (1986). Success runs and longest runs. Statist. Prob. Lett. 4, 211–215.Google Scholar
Rényi, A. (1970). Probability Theory. Academic Kiadó, Budapest.Google Scholar
Seneta, E. (1981). Nonnegative Matrices and Markov chains, 2nd edn. Springer, New York.Google Scholar
Schilling, M. F. (1990). The longest run of heads. College Math. J. 21, 196–207.Google Scholar
Suman, K. A. (1994). The longest run of any letter in a randomly generated word. In Runs and Patterns in Probability: Selected Papers (Math. Appl. 283), eds Godbole, A. P. and Papastavridis, S. G., Kluwer, Dordrecht, pp. 119–130.Google Scholar