Abstract
Let {Z t ,t≥1} be a sequence of trials taking values in a given setA={0, 1, 2,...,m}, where we regard the value 0 as failure and the remainingm values as successes. Let ε be a (single or compound) pattern. In this paper, we provide a unified approach for the study of two joint distributions, i.e., the joint distribution of the numberX n of occurrences of ε, the numbers of successes and failures inn trials and the joint distribution of the waiting timeT r until ther-th occurrence of ε, the numbers of successes and failures appeared at that time. We also investigate some distributions as by-products of the two joint distributions. Our methodology is based on two types of the random variablesX n (a Markov chain imbeddable variable of binomial type and a Markov chain imbeddable variable of returnable type). The present work develops several variations of the Markov chain imbedding method and enables us to deal with the variety of applications in different fields. Finally, we discuss several practical examples of our results.
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Aki, S. and Hirano, K. (1993). Discrete distributions related to succession events in a two-state Markov chain,Statistical Sciences and Data Analysis; Proceedings of the Third Pacific Area Statistical Conference (eds. K. Matusita, M. L. Puri and T. Hayakawa), 467–474, VSP International Science Publishers, Zeist.
Aki, S. and Hirano, K. (1994). Distributions of numbers of failures and successes until the first consecutivek successes,Annals of the Institute of Statistical Mathematics,46, 193–202.
Aki, S. and Hirano, K. (1995). Joint distributions of numbers of success-runs and failures until the first consecutivek successes,Annals of the Institute of Statistical Mathematics,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, Annals of the Institute of Statistical Mathematics,52, 767–777.
Aki, S., Balakrishnan, N. and Mohanty, S. G. (1996). Sooner and later waiting time problems and failure runs in higher order Markov dependent trials,Annals of the Institute of Statistical Mathematics,48, 773–787.
Balakrishnan, N. and Koutras, M. V. (2002).Runs and Scans with Applications, Wiley, New York.
Biggins, J. D. and Cannings, C. (1987). Markov renewal processes, counters and repeated sequences in Markov chains,Advances in Applied Probability,19, 521–545.
Blom, G. and Thorburn, D. (1982). How many random digits are required until given sequences are obtained?,Journal of Applied Probability,19, 518–531.
Chadjiconstantinidis, S., Antzoulakos, D. L. and Koutras, M. V. (2000). Joint distributions of successes, failures and patterns in enumeration problems,Advances in Applied Probability,32, 866–884.
Doi, M. and Yamamoto, E. (1998). On the joint distribution of runs in a sequence of multi-state trials,Statistics & Probability Letters,39, 133–141.
Ebneshahrashoob, M. and Sobel, M. (1990). Sooner and later waiting time problems for Bernoulli trials: Frequency and run quotas,Statistics & Probability Letters,9, 5–11.
Ewens, W. J. and Grant, G. R. (2001).Statistical Methods in Bioinformatics: An Introduction, Springer, New York.
Feller, W. (1968).An Introduction to Probability Theory and Its Applications, Vol. I, 3rd ed, Wiley, New York.
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. and Koutras, M. V. (1994). Distribution theory of runs: A Markov chain approach,Journal of the American Statistical Association,89, 1050–1058.
Fu, J. C. and Lou, W. Y. W. (2000). Joint distribution of rises and falls,Annals of the Institute of Statistical Mathematics,52, 415–425.
Glaz, J., Naus, J. and Wallenstein, S. (2001).Scan Statistics, Springer, New York.
Han, Q. and Aki, S. (1999). Joint distributions of runs in a sequence of multi-state trials,Annals of the Institute of Statistical Mathematics,51, 419–447.
Inoue, K. and Aki, S. (2002). Generalized waiting time problems associated with pattern in Polya’s urn scheme,Annals of the Institute of Statistical Mathematics,54, 681–688.
Inoue, K. and Aki, S. (2003). Generalized binomial and negative binomial distributions of orderk by the ℓ-overlapping enumeration scheme,Annals of the Institute of Statistical Mathematics,55, 153–167.
Koutras, M. V. (1996a). On a Markov chain approach for the study of reliability structures,Journal of Applied Probability,33, 357–367.
Koutras, M. V. (1996b). On a waiting time distribution in a sequence of Bernoulli trials,Annals of the Institute of Statistical Mathematics,48, 789–806.
Koutras, M. V. (1997). Waiting time distributions associated with runs of fixed length in two-state Markov chain,Annals of the Institute of Statistical Mathematics,49, 123–139.
Koutras, M. V. and 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.
Koutras, M. V. and Alexandrou, V. A. (1997a). Non-parametric randomness tests based on success runs of fixed length,Statistics & Probability Letters,32, 393–404.
Koutras, M. V. and Alexandrou, V. A. (1997b). Sooner waiting time problems in a sequence of trinary trials,Journal of Applied Probability,34, 593–609.
Mood, A. M. (1940). The distribution theory of runs,Annals of Mathematical Statistics,11, 367–392.
Philippou, A. N., Georghiou, C., and Philippou, G. N. (1983). A generalized geometric distribution and some of its properties,Statistics & Probability Letters,1, 171–175.
Robin, S. and Daudin, J. J. (1999). Exact distribution of word occurrences in a random sequence of letters,Journal of Applied Probability,36, 179–193.
Stefanov, V. T. (2000). On some waiting time problems.Journal of Applied Probability,37, 756–764.
Stefanov, V. T. and Pakes, A. G. (1997). Explicit distributional results in pattern formation,The Annals of Applied Probability,7, 666–678.
Stefanov, V. T. and Pakes, A. G., (1999). Explict distributional results in pattern formation II,Australian & New Zealand Journal of Statistics,41, 79–90.
Uchida, M. (1998). On generating functions of waiting time problems for sequence patterns of discrete random variables,Annals of the Institute of Statistical Mathematics,50, 655–671.
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This research was partially supported by the ISM Cooperative Research Program (2002-ISM·CRP-2007).
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Inoue, K. Joint distributions associated with patterns, successes and failures in a sequence of multi-state trials. Ann Inst Stat Math 56, 143–168 (2004). https://doi.org/10.1007/BF02530529
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DOI: https://doi.org/10.1007/BF02530529