Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-27T21:17:19.405Z Has data issue: false hasContentIssue false
  • English
  • Français

On probability generating functions for waiting time distributions of compound patterns in a sequence of multistate trials

Part of: Markov processes Distribution theory - Probability

Published online by Cambridge University Press:  14 July 2016

James C. Fu  and
Y. M. Chang
James C. Fu*
Affiliation:
University of Manitoba
Y. M. Chang*
Affiliation:
University of Manitoba
*
∗ Postal address: Department of Statistics, University of Manitoba, Winnipeg, Manitoba, Canada.
∗ Postal address: Department of Statistics, University of Manitoba, Winnipeg, Manitoba, Canada.
Get access Rights & Permissions [Opens in a new window]

Abstract

Probability generation functions of waiting time distributions of runs and patterns have been used successfully in various areas of statistics and applied probability. In this paper, we provide a simple way to obtain the probability generating functions for waiting time distributions of compound patterns by using the finite Markov chain imbedding method. We also study the characters of waiting time distributions for compound patterns. A computer algorithm based on Markov chain imbedding technique has been developed for automatically computing the distribution, probability generating function, and mean of waiting time for a compound pattern.

Keywords

Probability generating functioncompound patternMarkov chain imbeddingtransition probability matrix

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 39 , Issue 1 , March 2002 , pp. 70 - 80
Copyright
Copyright © Applied Probability Trust 2002 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

This work was supported in part by Nature Science and Engineering Research Council of Canada, under Grant A-9216.

References

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.CrossRefGoogle 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. Methodol. 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.CrossRefGoogle 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.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
Feller, W. (1968). An Introduction to Probability Theory and Its Applications, Vol. 1, 3rd edn. John Wiley, New York.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.Google Scholar
Fu, J. C. (1996). Distribution theory of runs and patterns associated with a sequence of multi-state trials. Statistica Sinica 6, 957–974.Google Scholar
Fu, J. C., and Koutras, M. V. (1994). Distribution theory of runs: A Markov chain approach. J. Amer. Statist. Assoc. 89, 1050–1058.CrossRefGoogle Scholar
Han, Q., and Aki, S. (1999). Joint distributions of runs in a sequence of multi-state trials. Ann. Inst. Statist. Math. 51, 419–447.CrossRefGoogle 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
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