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On Success Runs of Length Exceeded a Threshold

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  • Frosso S. Makri

    (University of Patras)

  • Zaharias M. Psillakis

    (University of Patras)

Abstract

Consider a sequence of n two state (success-failure) trials with outcomes arranged on a line or on a circle. The elements of the sequence are independent (identical or non identical distributed), exchangeable or first-order Markov dependent (homogeneous or non homogeneous) random variables. The statistic denoting the number of success runs of length at least equal to a specific length (a threshold) is considered. Exact formulae, lower/upper bounds and approximations are obtained for its probability distribution. The mean value and the variance of it are derived in an exact form. The distributions and the means of an associated waiting time and the length of the longest success run are provided. The reliability function of certain general consecutive systems is deduced using specific probabilities of the studied statistic. Detailed application case studies, covering a wide variety of fields, are combined with extensive numerical experimentation to illustrate further the theoretical results.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:metcap:v:13:y:2011:i:2:d:10.1007_s11009-009-9147-1
    DOI: 10.1007/s11009-009-9147-1
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    References listed on IDEAS

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    Cited by:

    1. Spiros D. Dafnis & Frosso S. Makri, 2023. "Distributions Related to Weak Runs With a Minimum and a Maximum Number of Successes: A Unified Approach," Methodology and Computing in Applied Probability, Springer, vol. 25(1), pages 1-24, March.
    2. Spiros D. Dafnis & Frosso S. Makri & Markos V. Koutras, 2021. "Generalizations of Runs and Patterns Distributions for Sequences of Binary Trials," Methodology and Computing in Applied Probability, Springer, vol. 23(1), pages 165-185, March.
    3. K. K. Kamalja, 2017. "Markov binomial distribution of order k and its applications," Statistical Papers, Springer, vol. 58(3), pages 831-853, September.
    4. Eryilmaz, Serkan, 2018. "On success runs in a sequence of dependent trials with a change point," Statistics & Probability Letters, Elsevier, vol. 132(C), pages 91-98.
    5. Eryilmaz, Serkan, 2015. "Discrete time shock models involving runs," Statistics & Probability Letters, Elsevier, vol. 107(C), pages 93-100.
    6. 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.
    7. 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.
    8. Vasileios M. Koutras & Markos V. Koutras & Spiros D. Dafnis, 2022. "A Family of Induced Distributions," Methodology and Computing in Applied Probability, Springer, vol. 24(3), pages 1833-1848, September.
    9. 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.

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