Compound Poisson law generalized by negative binomial distribution

Abstract

519.233.24 For the compound Poisson law generalized by the negative binomial distribution we derive the explicit representation of the probability function, the finite-difference recurrence, and expressions for the derivatives with respect to all parameters. Explicit representations of ordinary and factorial cumulants to sixth order are given and asymptotic normality of the distribution is proved. The distribution functions are constructed and analyzed for the most typical parameter values. Used to solve direct and inverse problems by moment methods.

Some properties of Neuman distribution of order k

“…Mixtures of the Poisson distribution of order k with binomial, negative binomial, and Poisson distributions have been considered in [3]. However, the presentation in [3] is strictly elementary and introductory, and the need for deeper analysis is suggested by our investigations of order-k Poisson -binomial [4,5] and order-k Poisson -negative binomial [6] distributions. In the present article we investigate some properties of a compound distribution involving a mixture of order-k Poisson with Poisson, i.e., a Neuman-A distribution of order k.…”