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Distributions Related to Weak Runs With a Minimum and a Maximum Number of Successes: A Unified Approach

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  • Spiros D. Dafnis

    (Agricultural University of Athens)

  • Frosso S. Makri

    (University of Patras)

Abstract

In the present paper we extend the definition of r-weak runs in sequences of binary trials so that such a run contains both a minimum and a maximum number of successes and we study the distribution of the statistic enumerating such (r-weak) runs. We are also investigating the distribution of the total number of successes in all the (r-weak) runs that are enumerated. The new introduced distributions may be of great applicability in scientific fields such as Agriculture. Our study requires an appropriate generalization of the Markov chain imbedding technique. More specifically, we introduce and study the family of Markov chain imbeddable variables of returnable-polynomial type, which generalizes the families of Markov chain imbeddable variables of binomial type, returnable type and polynomial type. The new family allows a unified approach for the study of the distributions of the statistics of current interest and other binomial-type distributions related to weak runs and typical runs.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:metcap:v:25:y:2023:i:1:d:10.1007_s11009-023-09998-x
    DOI: 10.1007/s11009-023-09998-x
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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. Muselli, Marco, 1996. "Simple expressions for success run distributions in bernoulli trials," Statistics & Probability Letters, Elsevier, vol. 31(2), pages 121-128, December.
    3. Serkan Eryilmaz, 2005. "On the distribution and expectation of success runs in nonhomogeneous Markov dependent trials," Statistical Papers, Springer, vol. 46(1), pages 117-128, January.
    4. 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.
    5. Spiros D. Dafnis & Frosso S. Makri, 2022. "Weak runs in sequences of binary trials," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 85(5), pages 573-603, July.
    6. Qing Han & Sigeo Aki, 1999. "Joint Distributions of Runs in a Sequence of Multi-State Trials," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 51(3), pages 419-447, September.
    7. 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.
    8. D. Antzoulakos & S. Bersimis & M. Koutras, 2003. "On the distribution of the total number of run lengths," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 55(4), pages 865-884, December.
    9. 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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