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

  • James C. Fu (a1) and Yung-Ming Chang (a2)

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.

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Corresponding author

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

References

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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. 385400.
Aki, S., and Hirano, K. (1999). Sooner and later waiting time problems for runs in Markov dependent bivariate trials. Ann. Inst. Statist. Math. 51, 1729.
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, 773787.
Balasubramanian, K., Viveros, R., and Balakrishnan, N. (1993). Sooner and later waiting time problems for Markovian Bernoulli trials. Statist. Prob. Lett. 18, 153161.
Boutsikas, M. V., and Koutras, M. V. (2000). Reliability approximation for Markov chain imbeddable systems. Methodology Comput. Appl. Prob. 2, 393411.
Chao, M. T., and Fu, J. C. (1989). A limit theorem for certain repairable systems. Ann. Inst. Statist. Math. 41, 809818.
Chao, M. T., and Fu, J. C. (1991). The reliability of a large series system under Markov structure. Adv. Appl. Prob. 23, 894908.
Doi, M., and Yamamoto, E. (1998). On the joint distribution of runs in a sequence of multistate trials. Statist. Prob. Lett. 39, 133141.
Ebneshahrashoob, M., and Sobel, M. (1990). Sooner and later problems for Bernoulli trials: frequency and run quotas. Statist. Prob. Lett. 9, 511.
Fu, J. C. (1986). Reliability of consecutive-k-out-of-n:F systems with (k-1)-step Markov dependence. IEEE Trans. Reliab. 35, 602606.
Fu, J. C. (1996). Distribution theory of runs and patterns associated with a sequence of multistate trials. Statistica Sinica 6, 957974.
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, 7080.
Fu, J. C., and Koutras, M. V. (1994). Distribution theory of runs: a Markov chain approach. J. Amer. Statist. Assoc. 89, 10501058.
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.
Koutras, M. V. (1997). Waiting time distributions associated with runs of fixed length in two-state Markov chains. Ann. Inst. Statist. Math. 49, 123139.
Koutras, M. V., and Alexandrou, V. A. (1997). Sooner waiting time problems in a sequence of trinary trials. J. Appl. Prob. 34, 593609.
Lou, W. Y. W. (1996). On runs and longest run tests: a method of finite Markov chain imbedding. J. Amer. Statist. Assoc. 91, 15951601.
Lou, W. Y. W. (1997). An application of the method of finite Markov chain imbedding to runs tests. Statist. Prob. Lett. 31, 155161.
Uchida, M. (1998). On generating functions of waiting time problems for sequence patterns of discrete random variables. Ann. Inst. Statist. Math. 50, 655671.

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

  • James C. Fu (a1) and Yung-Ming Chang (a2)

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