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Distribution of waiting time until the rth occurrence of a compound pattern

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  • Chang, Yung-Ming

Abstract

In this article, we investigate the waiting time distributions for the rth (r[greater-or-equal, slanted]1) occurrence of a compound pattern with respect to nonoverlapping and overlapping counting schemes in a sequence of Markov dependent multistate trials. Based on the finite Markov chain imbedding approach, the exact distribution, mean, variance and probability generating function are obtained. In addition, an alternative way of finding the probability generating function is presented.

Suggested Citation

  • Chang, Yung-Ming, 2005. "Distribution of waiting time until the rth occurrence of a compound pattern," Statistics & Probability Letters, Elsevier, vol. 75(1), pages 29-38, November.
    • Handle: RePEc:eee:stapro:v:75:y:2005:i:1:p:29-38
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    1. Sigeo Aki & Katuomi Hirano, 2000. "Numbers 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, Springer;The Institute of Statistical Mathematics, vol. 52(4), pages 767-777, December.
    2. S. Aki & N. Balakrishnan & S. Mohanty, 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, Springer;The Institute of Statistical Mathematics, vol. 48(4), pages 773-787, December.
    3. Sigeo Aki & Katuomi Hirano, 1999. "Sooner and Later Waiting Time Problems for Runs in Markov Dependent Bivariate Trials," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 51(1), pages 17-29, March.
    4. Han, Qing & Hirano, Katuomi, 2003. "Waiting time problem for an almost perfect match," Statistics & Probability Letters, Elsevier, vol. 65(1), pages 39-49, October.
    5. Ebneshahrashoob, M. & Sobel, Milton, 1990. "Sooner and later waiting time problems for Bernoulli trials: frequency and run quotas," Statistics & Probability Letters, Elsevier, vol. 9(1), pages 5-11, January.
    6. Balasubramanian, K. & Viveros, R. & Balakrishnan, N., 1993. "Sooner and later waiting time problems for Markovian Bernoulli trials," Statistics & Probability Letters, Elsevier, vol. 18(2), pages 153-161, September.
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    Cited by:

    1. Sankaran, P.G. & Midhu, N.N., 2010. "On waiting time distributions for patterns in a sequence of multistate trials," Statistics & Probability Letters, Elsevier, vol. 80(23-24), pages 1798-1805, December.
    2. Pozdnyakov, Vladimir, 2008. "On occurrence of patterns in Markov chains: Method of gambling teams," Statistics & Probability Letters, Elsevier, vol. 78(16), pages 2762-2767, November.
    3. Yu-Fei Hsieh & Tung-Lung Wu, 2013. "Recursive equations in finite Markov chain imbedding," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 65(3), pages 513-527, June.
    4. Bara Kim & Jeongsim Kim & Jerim Kim, 2020. "Waiting Time Problems for Patterns in a Sequence of Multi-State Trials," Mathematics, MDPI, vol. 8(11), pages 1-16, October.
    5. James C. Fu & W. Y. Wendy Lou, 2007. "On the Normal Approximation for the Distribution of the Number of Simple or Compound Patterns in a Random Sequence of Multi-state Trials," Methodology and Computing in Applied Probability, Springer, vol. 9(2), pages 195-205, June.
    6. Yung-Ming Chang & James Fu & Han-Ying Lin, 2012. "Distribution and double generating function of number of patterns in a sequence of Markov dependent multistate trials," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 64(1), pages 55-68, February.

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