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