Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-26T20:27:03.141Z Has data issue: false hasContentIssue false
  • English
  • Français

Sooner waiting time problems in a sequence of trinary trials

Part of: Markov processes Distribution theory - Probability

Published online by Cambridge University Press:  14 July 2016

M. V. Koutras  and
V. A. Alexandrou
M. V. Koutras*
Affiliation:
University of Athens
V. A. Alexandrou*
Affiliation:
University of Athens
*
∗Postal address: Department of Mathematics, University of Athens, Panepistemiopolis, 15784, Greece.
∗Postal address: Department of Mathematics, University of Athens, Panepistemiopolis, 15784, Greece.
Get access Rights & Permissions [Opens in a new window]

Abstract

Let X1, X2,· ·· be a (linear or circular) sequence of trials with three possible outcomes (say S, S∗ or F) in each trial. In this paper, the waiting time for the first appearance of an S-run of length k or an S∗-run of length r is systematically investigated. Exact formulae and Chen-Stein approximations are derived for the distribution of the waiting times in both linear and circular problems and their asymptotic behaviour is illustrated. Probability generating functions are also obtained when the trials are identical. Finally, practical applications of these results are discussed in some detail.

Keywords

SUCCESS RUNSSOONER WAITING TIMESMULTISTATE TRIALSMARKOV CHAINSCHEN-STEIN APPROXIMATIONS

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 34 , Issue 3 , September 1997 , pp. 593 - 609
Copyright
Copyright © Applied Probability Trust 1997 

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.)

References

AKi, S. (1992) Waiting time problems for a sequence of discrete random variables. Ann. Inst. Statist. Math. 44, 363–378.CrossRefGoogle Scholar
Aki, S. and Hirano, K. (1993) Discrete distributions related to succession events in a two-state Markov chain. In Statistical Science and Data Analysis. ed. Matusita, K. et al. VSP, Amsterdam. pp. 467–474.CrossRefGoogle Scholar
Alevizos, P., Papastavridis, S. and Sypsas, P. (1993) Reliability of cyclic m-consecutive-k-out-of-n:F systems. In Reliability Control and Risk Assessment. Proc. IASTED Conf. pp. 140–143.Google Scholar
Arratia, R., Goldstein, L. and Gordon, L. (1989) Two moments suffice for Poisson approximations: the Chen-Stein method. Ann. Prob. 17, 9–25.CrossRefGoogle Scholar
Balakrishnan, N., Balasubramanian, K. and Viveros, R. (1993) On sampling inspection plans based on the theory of runs. Math. Sci. 18, 113–126.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.CrossRefGoogle Scholar
Barbour, A. D., Holst, L. and Janson, S. (1992) Poisson Approximation. Oxford University Press, Oxford.CrossRefGoogle Scholar
Bogartz, R. (1965) The criterion method: some analyses and remarks. Psychol. Bull. 64, 1–14.CrossRefGoogle ScholarPubMed
Bradley, J. (1968) Distribution Free Statistical Tests. Prentice-Hall, New Jersey.Google Scholar
Chryssaphinou, O., Papastavridis, S. and Tsapelas, T. (1994) On the waiting time of appearance of given patterns. In Runs and Patterns in Probability. ed. Godbole, A. and Papastavridis, S. Kluwer, Dordrecht. pp. 231–241.CrossRefGoogle Scholar
Ebneshahrashoob, M. and Sobel, M. (1990) Sooner and later waiting time problems for Bernoulli trials: frequency and run quotas. Statist. Prob. Lett. 9, 5–11.CrossRefGoogle Scholar
Feller, W. (1968) An Introduction to Probability Theory and Its Applications. Vol. I. 3rd edn. Wiley, New York.Google Scholar
Fu, J.C. (1995) Exact and limiting distributions of the number of successions in a random permutation. Ann. Inst. Statist. Math. 47, 435–446.CrossRefGoogle Scholar
Fu, J. C. (1996) Distribution theory of runs and patterns associated with a sequence of multistate trials. Statist. 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
Grant, D. (1946) New statistical criteria for learning and problem solution in experiments involving repeated trials. Psychol. Bull. 43, 272–282.CrossRefGoogle ScholarPubMed
Hirano, K. and Aki, S. (1993) On number of occurrences of success runs of specified length in a two-state Markov chain. Statist. Sinica 3, 313–320.Google Scholar
Koutras, M. V. and Alexandrou, V. A. (1995) Runs, scans and urn model distributions: A unified Markov chain approach. Ann. Inst. Statist. Math. 47, 743–766.CrossRefGoogle Scholar
Koutras, M. V., Papadopoulos, G. K. and Papastavridis, S. G. (1994) Circular overlapping success runs. In Runs and Patterns in Probability. ed. Godbole, A. and Papastavridis, S. Kluwer, Dordrecht. pp. 287–305.CrossRefGoogle Scholar
Koutras, M. V., Papadopoulos, G. K. and Papastavridis, S. G. (1995) Runs on a circle. J. Appl. Prob. 32, 396–404.CrossRefGoogle Scholar
Koutras, M. V. and Papastavridis, S. G. (1993) On the number of runs and related statistics. Statist. Sinica 3, 277–294.Google Scholar
Ling, K. D. (1990) On geometric distributions of order (k1,, km) . Statist. Prob. Lett. 9, 163–171.CrossRefGoogle Scholar
Ling, K. and Low, T. (1993) On the soonest and latest waiting time distributions: succession quotas. Commun. Statist.-Theory Meth. 22, 2207–2221.CrossRefGoogle Scholar
Makri, F. S. and Philippou, A. N. (1994) Binomial distributions of order k on the circle. In Runs and Patterns in Probability. ed. Godbole, A. and Papastavridis, S. Kluwer, Dordrecht. pp. 65–81.CrossRefGoogle Scholar
Prairie, R. R., Zimmer, W. J. and Brookhouse, J. K. (1962) Some acceptance sampling plans based on the theory of runs. Technometrics 4, 177–185.CrossRefGoogle Scholar
Rajarshi, M. B. (1974) Success runs in a two-state Markov chain. J. Appl. Prob. 11, 190–194.CrossRefGoogle Scholar
Viveros, R. and Balakrishnan, N. (1993) Statistical inference from start-up demonstration test data. J. Qual. Technol. 22, 119–130.CrossRefGoogle Scholar