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On the Distribution of the Number of Success Runs in a Continuous Time Markov Chain

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  • Boutsikas V. Michael

    (University of Piraeus)

  • Vaggelatou Eutichia

    (National and Kapodistrian University of Athens)

Abstract

We propose a continuous-time adaptation of the well-known concept of success runs by considering a marked point process with two types of marks (success-failure) that appear according to an appropriate continuous-time Markov chain. By constructing a bivariate imbedded process (consisting of a run-counting and a phase process), we offer recursive formulas and generating functions for the distribution of the number of runs and the waiting time until the appearance of the n-th success run. We investigate the three most popular counting schemes: (i) overlapping runs of length k, (ii) non-overlapping runs of length k and (iii) runs of length at least k. We also present examples of applications regarding: the total penalty cost in a maintenance reliability system, the number of risky situations in a non-life insurance portfolio and the number of runs of increasing (or decreasing) asset price movements in high-frequency financial data.

Suggested Citation

  • Boutsikas V. Michael & Vaggelatou Eutichia, 2020. "On the Distribution of the Number of Success Runs in a Continuous Time Markov Chain," Methodology and Computing in Applied Probability, Springer, vol. 22(3), pages 969-993, September.
    • Handle: RePEc:spr:metcap:v:22:y:2020:i:3:d:10.1007_s11009-019-09743-3
    DOI: 10.1007/s11009-019-09743-3
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    References listed on IDEAS

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    1. Frosso S. Makri & Zaharias M. Psillakis, 2011. "On Success Runs of Length Exceeded a Threshold," Methodology and Computing in Applied Probability, Springer, vol. 13(2), pages 269-305, June.
    2. Frosso S. Makri & Zaharias M. Psillakis, 2016. "On runs of ones defined on a q-sequence of binary trials," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 79(5), pages 579-602, July.
    3. Wu, Tung-Lung & Glaz, Joseph, 2015. "A new adaptive procedure for multiple window scan statistics," Computational Statistics & Data Analysis, Elsevier, vol. 82(C), pages 164-172.
    4. Serkan Eryilmaz, 2018. "Stochastic Ordering Among Success Runs Statistics in a Sequence of Exchangeable Binary Trials," Methodology and Computing in Applied Probability, Springer, vol. 20(2), pages 563-573, June.
    5. Binswanger, K. & Embrechts, P., 1994. "Longest runs in coin tossing," Insurance: Mathematics and Economics, Elsevier, vol. 15(2-3), pages 139-149, December.
    6. Boutsikas, M. V. & Koutras, M. V., 2002. "Modeling claim exceedances over thresholds," Insurance: Mathematics and Economics, Elsevier, vol. 30(1), pages 67-83, February.
    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. He Yi & Lirong Cui & Narayanaswamy Balakrishnan, 2022. "On the Derivative Counting Processes of First- and Second-order Aggregated Semi-Markov Systems," Methodology and Computing in Applied Probability, Springer, vol. 24(3), pages 1849-1875, September.

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