Abstract
In this paper, we investigate the exact distribution of the waiting time for ther-th ℓ-overlapping occurrence of success-runs of a specified length in a sequence of two state Markov dependent trials. The probability generating functions are derived explicitly, and as asymptotic results, relationships of a negative binomial distribution of orderk and an extended Poisson distribution of orderk are discussed. We provide further insights into the run-related problems from the viewpoint of the ℓ-overlapping enumeration scheme. We also study the exact distribution of the number of ℓ-overlapping occurrences of success-runs in a fixed number of trials and derive the probability generating functions. The present work extends several properties of distributions of orderk and leads us a new type of geneses of the discrete 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. (1985). Discrete distributions of orderk on a binary sequence,Ann. Inst. Statist. Math.,37 205–224.
Aki, S. and Hirano, K. (1988). Some characteristics of the binomial distribution of orderk and related distributions,Statistical Theory and Data Analysis II, Proceedings of the Second Pacific Area Statistical Conference (eds. K. Matusita), 211–222, North-Holland, Amsterdam.
Aki, S. and Hirano, K. (1994). Distributions of numbers of failures and successes until the first consectivek successes,Ann. Inst. Statist. Math.,46, 193–202.
Aki, S. and Hirano, K. (1995). Joint distributions of numbers of success- runs and failures until the first consecutivek successes,Ann. Inst. Statist. Math.,47, 225–235.
Aki, S. and Hirano, K. (2000). Numbers of success-runs of specified length until certain stopping time rules and generalized binomial distributions of orderk, Ann. Inst. Statist. Math.,52, 767–777.
Aki, S., Kuboki, H. and Hirano, K. (1984). On discrete distributions of orderk, Ann. Inst. Stat Math.,36, 431–440.
Antozoulakos, D. L. (1999). On waiting time problems associated with runs in Markov dependent trials,Ann. Inst. Statist. Math.,51, 323–330.
Balakrishnan, N. and Koutras, M. V. (2002).Runs and Scans with Applications, Wiley, New York.
Chadjiconstantinidis, S., Antzoulakos, D. L. and Koutras, M. V. (2000). Joint distributions of successes, failures and patterns in enumeration problems,Adv. in Appl. Probab.,32, 866–884.
Chao, M. T. (1991). The reliability of large series systems under Markov structure,Adv. in Appl. Probab.,23, 894–908.
Chao, M. T. and Fu, J. C. (1989). A limit theorem of certain repairable systems,Ann. Inst. Statist. Math.,41, 809–818.
Feller, W. (1968).An Introduction to Probability Theory and Its Applications, Vol. I, 3rd ed., Wiley, New York.
Fu, J. C. (1986). Reliability of consecutive-k-out-of-n: F systems with (k−1)-step Markov dependence,IEEE Transactions on Reliability,R35, 602–606.
Fu, J. C. and Hu, B. (1987). On reliability of a large consecutive-k-out-of-n: F system with (k−1)-step Markov dependence,IEEE Transactions on Reliability,R36, 75–77.
Fu, J. C. and Koutras, M. V. (1994). Distribution theory of runs: A Markov chain approach,J. Amer. Statist. Assoc.,89, 1050–1058.
Han, Q. and Aki, S. (1998). Formulae and recursions for the joint distributions of success runs of several lengths in a two-state Markov chain,Statist. Probab. Lett.,40, 203–214.
Han, S. and Aki, S. (2000). A unified approach to binomial-type distributions of orderk, Comm. Statist. Theory Methods,29(8), 1929–1943.
Hirano, K. and Aki, S. (1987). Properties of the extended distributions of orderk, Statist. Probab. Lett.,6, 67–69.
Hirano, K., Aki, S., Kashiwagi, N. and Kuboki, H. (1991). On Ling’s binomial and negative binomial distributions of orderk, Statist. Probab. Lett.,11, 503–509.
Koutras, M. V. (1996a). On a Markov chain approach for the study of reliability structures,J. Appl. Probab.,33, 357–367.
Koutras, M. V. (1996b). On a waiting time distribution in a sequence of Bernoulli trials,Ann. Inst. Statist. Math.,48, 789–806.
Koutras, M. V. (1997). Waiting time distributions associated with runs of fixed length in two- state Markov chain,Ann. Inst. Statist. Math.,49, 123–139.
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.
Koutras, M. V. and Alexandrou, V. A. (1997). Sooner waiting time problems in a sequence of trinary trials,J. Appl. Probab.,34, 593–609.
Koutras, M. V., Papadopoulos, G. K. and Papastavridis, S. G. (1995). Runs on a circle,J. Appl. Probab.,32, 396–404.
Ling, K. D. (1988). On binomial distributions of orderk, Statist. Probab. Lett.,6, 247–250.
Philippou, A. N. (1988). On multiparameter distributions of orderk, Ann. Inst. Statist. Math.,40, 467–475.
Philippou, A. N., Georghiou, C. and Philippou, G. N. (1983). A generalized geometric distribution and some of its properties,Statist. Probab. Lett.,1, 171–175.
Uchida, M. (1998). On number of occurrences of success runs of specified length in a higher-order two state Markov chain,Ann. Inst. Statist. Math.,50, 587–601.
Zhang, F. Z. (1999).Matrix Theory, Springer, New York.
Author information
Authors and Affiliations
About this article
Cite this article
Inoue, K., Aki, S. Generalized binomial and negative binomial distributions of orderk by thel-overlapping enumeration scheme. Ann Inst Stat Math 55, 153–167 (2003). https://doi.org/10.1007/BF02530491
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02530491