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On ordered series and later waiting time distributions in a sequence of Markov dependent multistate trials

Part of: Distribution theory - Probability Markov processes

Published online by Cambridge University Press:  14 July 2016

James C. Fu  and
Yung-Ming Chang
James C. Fu*
Affiliation:
University of Manitoba
Yung-Ming Chang*
Affiliation:
National University of Kaohsiung
*
∗ Postal address: Department of Statistics, University of Manitoba, Winnipeg, Manitoba R3T 2N2, Canada. Email address: james_fu@umanitoba.ca
∗ Postal address: Department of Applied Mathematics, National University of Kaohsiung, Kaohsiung 811, Taiwan
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Abstract

The sooner and later waiting time problems have been extensively studied and applied in various areas of statistics and applied probability. In this paper, we give a comprehensive study of ordered series and later waiting time distributions of a number of simple patterns with respect to nonoverlapping and overlapping counting schemes in a sequence of Markov dependent multistate trials. Exact distributions and probability generating functions are derived by using the finite Markov chain imbedding technique. Examples are given to illustrate our results.

Keywords

Compound patternordered series patternlater waiting timeMarkov chain imbeddingtransition probability matrixprobability generating function

MSC classification

Primary: 60E05: Distributions
Secondary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)

Information

Type
Research Papers
Information
Journal of Applied Probability , Volume 40 , Issue 3 , September 2003 , pp. 623 - 642
Copyright
Copyright © Applied Probability Trust 2003 

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References

Aki, S. (1997). On sooner and later problems between success and failure runs. In Advances in Combinatorial Methods and Applications to Probability and Statistics, ed. Balakrishnan, N., Birkhäuser, Boston, MA, pp. 385–400.Google Scholar
Aki, S., and Hirano, K. (1999). Sooner and later waiting time problems for runs in Markov dependent bivariate trials. Ann. Inst. Statist. Math. 51, 17–29.Google Scholar
Aki, S., Balakrishnan, N., and Mohanty, S. G. (1996). Sooner and later waiting time problems for success and failure runs in higher order Markov dependent trials. Ann. Inst. Statist. Math. 48, 773–787.Google Scholar
Balasubramanian, K., Viveros, R., and Balakrishnan, N. (1993). Sooner and later waiting time problems for Markovian Bernoulli trials. Statist. Prob. Lett. 18, 153–161.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
Chao, M. T., and Fu, J. C. (1989). A limit theorem for 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 Markov structure. Adv. Appl. Prob. 23, 894–908.CrossRefGoogle Scholar
Doi, M., and Yamamoto, E. (1998). On the joint distribution of runs in a sequence of multistate trials. Statist. Prob. Lett. 39, 133–141.Google Scholar
Ebneshahrashoob, M., and Sobel, M. (1990). Sooner and later problems for Bernoulli trials: frequency and run quotas. Statist. Prob. Lett. 9, 5–11.Google Scholar
Fu, J. C. (1986). Reliability of consecutive-k-out-of-n:F systems with (k-1)-step Markov dependence. IEEE Trans. Reliab. 35, 602–606.CrossRefGoogle Scholar
Fu, J. C. (1996). Distribution theory of runs and patterns associated with a sequence of multistate trials. Statistica 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.CrossRefGoogle Scholar
Fu, J. C., and Koutras, M. V. (1994). Distribution theory of runs: a Markov chain approach. J. Amer. Statist. Assoc. 89, 1050–1058.Google Scholar
Fu, J. C., and Lou, W. Y. W. (2003). Distribution Theory of Runs and Patterns and Its Applications: A Finite Markov Chain Approach. World Scientific, Singapore.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. Assoc. 91, 1595–1601.CrossRefGoogle Scholar
Lou, W. Y. W. (1997). An application of the method of finite Markov chain imbedding to runs tests. Statist. Prob. Lett. 31, 155–161.Google Scholar
Uchida, M. (1998). On generating functions of waiting time problems for sequence patterns of discrete random variables. Ann. Inst. Statist. Math. 50, 655–671.CrossRefGoogle Scholar