Four Routes to Mixed Poisson Distributions

Abstract

The objective of this work is to express mixed Poisson distributions in four ways; namely, in explicit form, in terms of special functions, in recursive form and in terms of transforms also called expectation forms. In explicit form, a gamma function and its properties is used. Posterior distributions and posterior moments are also obtained. Modified Bessel function of the third kind and confluent hypergeometric function with their properties are used in expressing mixed Poisson distributions in terms of special functions. Integration by parts is used in determining recursive models for mixed Poisson distributions. To determine the corresponding differential equations for these recursive models, Wang’s recursive approach is applied. Laplace transform and jth moment of a mixing distribution are used to express Poisson mixtures in expectation forms. Factorial moments, moments about the origin and moments about the mean of the Poisson mixtures are determined in terms of probability generating functions of the mixtures. A major bottle-neck in using Laplace transform technique is to obtain its xth derivative. Determining some mathematical identities based on Poisson mixtures is a major contribution in this research. These identities are obtained by equating results derived using explicit forms and their corresponding method of moments. Identities are also obtained by equating Poisson mixtures expressed in terms of special functions and their corresponding method of moments. The other major contribution is use of integration by parts in determining recursive models. Other researchers obtained similar results but with certain conditions to be fulfilled. The integration by parts approach does not need these conditions. In literature, Lindley distribution has been generalized to two parameters. A contribution in this research work is the construction of a three-parameter generalized Lindley distribution which nests the one and two parameter Lindley distributions. The focus of this research is on constructions and properties of mixed Poisson distributions. For further research, estimations and applications could be pursued. Other approaches to constructing Poisson mixtures could also be identified and pursued