Generalized distributions of order \(k\) associated with success runs in Bernoulli trials (Q1864316)

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 \textit{G. C. Jain} and \textit{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 \textit{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.