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Formulae and recursions for the joint distributions of success runs of several lengths in a two-state Markov chain

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  • Han, Qing
  • Aki, Sigeo

Abstract

Let X1, X2,...,Xn be a time-homogeneous {0, 1}-valued Markov chain. Let Y = (Y1,...,Yr) denote the r-dimensional random vector, where Yi(i = l,...,r) represents the number of success runs of length ki(i = 1,...,r), k = (kl,...,kr). In this paper we obtain exact and recurrence formulae for the probability functions and the probability generating functions of Y, based on four different ways of counting numbers of success runs (i.e. overlapping success runs, non-overlapping runs, the runs with a specified length k or more and the runs with just specified length k). Using our methods, we can obtain the probability function and probability generating function of Y, where Y1,...,Yr be counted numbers by different ways.

Suggested Citation

  • Han, Qing & Aki, Sigeo, 1998. "Formulae and recursions for the joint distributions of success runs of several lengths in a two-state Markov chain," Statistics & Probability Letters, Elsevier, vol. 40(3), pages 203-214, October.
    • Handle: RePEc:eee:stapro:v:40:y:1998:i:3:p:203-214
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    References listed on IDEAS

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    1. Anant Godbole & Stavros Papastavridis & Robert Weishaar, 1997. "Formulae and Recursions for the Joint Distribution of Success Runs of Several Lengths," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 49(1), pages 141-153, March.
    2. Ling, K. D., 1988. "On binomial distributions of order k," Statistics & Probability Letters, Elsevier, vol. 6(4), pages 247-250, March.
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    5. Ling, K. D., 1989. "A new class of negative binomial distributions of order k," Statistics & Probability Letters, Elsevier, vol. 7(5), pages 371-376, April.
    6. Hirano, K. & Aki, S. & Kashiwagi, N. & Kuboki, H., 1991. "On Ling's binomial and negative binomial distributions of order k," Statistics & Probability Letters, Elsevier, vol. 11(6), pages 503-509, June.
    7. M. Koutras & V. Alexandrou, 1995. "Runs, scans and URN model distributions: A unified Markov chain approach," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 47(4), pages 743-766, December.
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    Cited by:

    1. Inoue, Kiyoshi & Aki, Sigeo, 2010. "On the conditional and unconditional distributions of the number of success runs on a circle with applications," Statistics & Probability Letters, Elsevier, vol. 80(9-10), pages 874-885, May.
    2. Kirtee Kamalja, 2014. "On the joint distribution of success runs of several lengths in the sequence of MBT and its applications," Statistical Papers, Springer, vol. 55(4), pages 1179-1206, November.
    3. Kiyoshi Inoue & Sigeo Aki, 2007. "Joint Distributions of Numbers of Runs of Specified Lengths in a Sequence of Markov Dependent Multistate Trials," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 59(3), pages 577-595, September.
    4. Kiyoshi Inoue & Sigeo Aki, 2018. "Joint distributions of numbers of runs of specified lengths on directed trees," Statistical Papers, Springer, vol. 59(1), pages 249-269, March.
    5. Bellini, Fabio & Figa-Talamanca, Gianna, 2005. "Runs tests for assessing volatility forecastability in financial time series," European Journal of Operational Research, Elsevier, vol. 163(1), pages 102-114, May.

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