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

Spiros D. Dafnis () and Frosso S. Makri
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Spiros D. Dafnis: Agricultural University of Athens
Frosso S. Makri: University of Patras

Methodology and Computing in Applied Probability, 2023, vol. 25, issue 1, 1-24

Abstract: 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.

Keywords: (weak) success runs; Run lengths; Markov chains; Markov chain imbeddable variables of returnable-polynomial type; Heat-unit models; 60E05; 60J10; 62E15 (search for similar items in EconPapers)
Date: 2023
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DOI: 10.1007/s11009-023-09998-x

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