Abstract
Consider a time homogeneous {0, 1}-valued m-dependent Markov chain \(\{X_{- m + 1 + n}, n \geqslant 0\}\). In this paper, we study the joint probability distribution of number of 0-runs of length \(k_{0} (k_{0} \geqslant m)\) and number of 1-runs of length \(k_{1} (k_{1} \geqslant m)\) in n trials. We study the joint distributions based on five popular counting schemes of runs. The main tool used to obtain the probability generating function of the joint distribution is the conditional probability generating function method. Further a compact method for the evaluation of exact joint distribution is developed. For higher-order two-state Markov chain, these joint distributions are new in the literature of distributions of run statistics. We use these distributions to derive some waiting time distributions.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Aki S. (1997). On sooner and later problems between success and failure runs. In: Balakrishnan N.(Eds), Advances in combinatorial methods and applications to probability and statistics. Boston, Birkhäuser, pp 401–410
Aki S., Hirano K. (1994). Distributions of numbers of failures and successes until the first consecutive k successes. Annals of the Institute of Statistical Mathematics, 46, 193–202
Aki S., Hirano K. (1995). Joint distributions of numbers of success-runs and failures until the first consecutive k successes. Annals of the Institute of Statistical Mathematics, 47, 225–235
Aki S., Hirano K. (2000). Number of success runs of specified length until certain stopping time rules and generalized binomial distributions of order k. Annals of the Institute of Statistical Mathematics, 52, 767–777
Aki S., Balakrishnan N., Mohanty S.G. (1996). Sooner and later waiting time problems for success and failure runs in higher-order Markov dependent trials. Annals of the Institute of Statistical Mathematics, 48, 773–787
Antzoulakos D.L. (1999). On waiting time problems associated with runs in Markov dependent trials. Annals of the Institute of Statistical Mathematics, 51, 323–330
Balakrishnan N. (1997). Joint distributions of numbers of success-runs and failures until the first consecutive k successes in a binary sequence. Annals of the Institute of Statistical Mathematics, 49, 519–529
Doi M., Yamamoto E. (1998). On the joint distribution of runs in a sequence of multi-state trials. Statistics & Probability Letters, 39, 133–141
Ebneshahrashoob M., Sobel M. (1990). Sooner and later waiting time problems for Bernoulli trials: frequency and run quotas. Statistics & Probability Letters, 9, 5–11
Feller W. (1968). An Introduction to probability theory and its applications, vol. I (3rd edn). New York, Wiley
Fu J.C. (1996). Distribution theory of runs and patterns associated with a sequence of multi-state trials. Statistica Sinica, 6, 957–974
Fu J.C., Koutras M.V. (1994). Distribution theory of runs: A Markov chain approach. Journal of American Statistical Association, 89, 1050–1058
Han Q., Aki S. (1999). Joint distributions of runs in a sequence of multi-state trials. Annals of the Institute of Statistical Mathematics, 51, 419–447
Han Q., Aki S. (2000a). Waiting time problems in a two-state Markov chain. Annals of the Institute of Statistical Mathematics, 52, 778–789
Han S., Aki S. (2000b). A unified approach to binomial-type distributions of order k. Communications in Statistics- Theory and Methods, 29(8): 1929–1943
Han Q., Aki S. (2000c). Sooner and later waiting time problems based on dependent sequence. Annals of the Institute of Statistical Mathematics, 52, 407–414
Hirano H., Aki S. (1993). On number of occurrences of success runs of specified length in a two-state Markov chain. Statistica Sinica, 3, 313–320
Inoue K., Aki S. (2002). Generalized waiting time problems associated with patterns in Polya’s urn scheme. Annals of the Institute of Statistical Mathematics, 54, 681–688
Inoue K., Aki S. (2003). Generalized binomial and negative binomial distributions of order k by the l-overlapping enumeration scheme. Annals of the Institute of Statistical Mathematics, 55, 153–167
Koutras M.V., Alexandrou V.A. (1995). Runs, scans and urn model distributions: A unified Markov chain approach. Annals of the Institute of Statistical Mathematics, 47, 743–766
Ling K.D. (1988). On binomial distributions of order k. Statistics & Probability Letters, 6, 247–250
Mood A.M. (1940). The distribution theory of runs. Annals of Mathematical Statistics, 11, 367–392
Schwager S.J. (1983). Run probability in sequences of Markov-dependent trials. Journal of American Statistical Association, 78, 168–175
Shinde R.L., Kotwal K.S. (2006). On the joint distribution of runs in the sequence of Markov dependent multi-state trials. Statistics and Probability letter, 76(10): 1065–1074
Uchida M. (1998). On number of occurrences of success runs of specified length in a higher-order two state Markov Chain. Annals of the Institute of Statistical Mathematics, 50, 587–601
Uchida M., Aki S. (1995). Sooner and later waiting time problems in a two-state Markov chain. Annals of the Institute of Statistical Mathematics, 47, 415–433
Author information
Authors and Affiliations
Corresponding author
About this article
Cite this article
Kotwal, K.S., Shinde, R.L. Joint distributions of runs in a sequence of higher-order two-state Markov trials. Ann Inst Stat Math 58, 537–554 (2006). https://doi.org/10.1007/s10463-005-0024-6
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10463-005-0024-6