Generalized distributions of order \(k\) associated with success runs in Bernoulli trials. (English) Zbl 1047.62012

Summary: In a sequence of independent Bernoulli trials, by counting multidimensional lattice paths in order to compute the probability of a first-passage event, we derive and study a generalized negative binomial distribution of order \(k\), type \(I\), which extends to distributions of order \(k\) the generalized negative binomial distribution of G. C. Jain and P. C. Consul [SIAM J. Appl. Math. 21, 501–513 (1971; Zbl 0234.60010)], and includes as a special case the negative binomial distribution of order \(k\), type \(I\), of A. N. Philippou et al. [Biom. J. 26, 789–794 (1984; Zbl 0566.60014); Stat. Probab. Lett. 7, 207–216 (1988; Zbl 0678.62058); ibid. 10, 29–35 (1990; Zbl 0716.62049)]. This new distribution gives rise in the limit to generalized logarithmic and Borel-Tanner distributions and, by compounding, to the generalized Pólya distribution of the same order and type. Limiting cases are considered and an application to observed data is presented.


62E10 Characterization and structure theory of statistical distributions
60E05 Probability distributions: general theory
60C05 Combinatorial probability
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