Generalized negative binomial distributions as mixed geometric laws and related limit theorems

TL;DR

This paper introduces and analyzes the generalized negative binomial distributions as mixtures of generalized gamma laws, explores their properties, and establishes limit theorems, with applications in meteorology.

Contribution

It provides a comprehensive study of GNB distributions, including their analytic properties, explicit mixing representations, and new limit theorems for sums involving GNB-distributed counts.

Findings

GNB distributions are shown to be mixed Poisson laws with explicit mixing distributions.

Limit theorems demonstrate convergence of mixed binomial to mixed Poisson laws.

Representations of limit laws involve mixtures of Mittag-Leffler, Linnik, or Laplace distributions.

Abstract

In this paper we study a wide and flexible family of discrete distributions, the so-called generalized negative binomial (GNB) distributions that are mixed Poisson distributions in which the mixing laws belong to the class of generalized gamma (GG) distributions. The latter was introduced by E. W. Stacy as a special family of lifetime distributions containing gamma, exponential power and Weibull distributions. These distributions seem to be very promising in the statistical description of many real phenomena being very convenient and almost universal models for the description of statistical regularities in discrete data. Analytic properties of GNB distributions are studied. A GG distribution is proved to be a mixed exponential distribution if and only if the shape and exponent power parameters are no greater than one. The mixing distribution is written out explicitly as a scale mixture of strictly stable laws concentrated on the nonnegative halfline. As a corollary, the representation is obtained for the GNB distribution as a mixed geometric distribution. The corresponding scheme of Bernoulli trials with random probability of success is considered. Within this scheme, a random analog of the Poisson theorem is proved establishing the convergence of mixed binomial distributions to mixed Poisson laws. Limit theorems are proved for random sums of independent random variables in which the number of summands has the GNB distribution and the summands have both light- and heavy-tailed distributions. The class of limit laws is wide enough and includes the so-called generalized variance gamma distributions. Various representations for the limit laws are obtained in terms of mixtures of Mittag-Leffler, Linnik or Laplace distributions. Some applications of GNB distributions in meteorology are discussed.