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Recursive equations in finite Markov chain imbedding

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  • Yu-Fei Hsieh
  • Tung-Lung Wu

Abstract

In this paper, recursive equations for waiting time distributions of r-th occurrence of a compound pattern are studied via the finite Markov chain imbedding technique under overlapping and non-overlapping counting schemes in sequences of independent and identically distributed (i.i.d.) or Markov dependent multi-state trials. Using the relationship between number of patterns and r-th waiting time, distributions of number of patterns can also be obtained. The probability generating functions are also obtained. Examples and numerical results are given to illustrate our theoretical results. Copyright The Institute of Statistical Mathematics, Tokyo 2013

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  • 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.
    • Handle: RePEc:spr:aistmt:v:65:y:2013:i:3:p:513-527
    DOI: 10.1007/s10463-012-0381-x
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    References listed on IDEAS

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    1. 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.
    2. 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.
    3. C. Fu, James & Shmueli, Galit & Chang, Y. M., 2003. "A unified Markov chain approach for computing the run length distribution in control charts with simple or compound rules," Statistics & Probability Letters, Elsevier, vol. 65(4), pages 457-466, December.
    4. Sigeo Aki, 1992. "Waiting time problems for a sequence of discrete random variables," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 44(2), pages 363-378, June.
    5. Kiyoshi Inoue & Sigeo Aki, 2005. "A generalized Pólya urn model and related multivariate distributions," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 57(1), pages 49-59, March.
    6. Kiyoshi Inoue & Sigeo Aki, 2009. "On waiting time distributions associated with compound patterns in a sequence of multi-state trials," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 61(2), pages 499-516, June.
    7. Fu, James C. & Lou, W. Y. Wendy & Wang, Yueh-Jir, 1999. "On the exact distributions of Eulerian and Simon Newcomb numbers associated with random permutations," Statistics & Probability Letters, Elsevier, vol. 42(2), pages 115-125, April.
    8. Yung-Ming Chang & Tung-Lung Wu, 2011. "On Average Run Lengths of Control Charts for Autocorrelated Processes," Methodology and Computing in Applied Probability, Springer, vol. 13(2), pages 419-431, June.
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    Cited by:

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    2. Wang, Xiaoyue & Zhao, Xian & Wang, Siqi & Sun, Leping, 2020. "Reliability and maintenance for performance-balanced systems operating in a shock environment," Reliability Engineering and System Safety, Elsevier, vol. 195(C).

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