Statistics Colloquium, Dr. John L Spouge

Title: An Exposition on Theoretical Topics Related to the Generalized Poisson Distribution (GPD)

Abstract: 

The GPD, like the Negative Binomial distribution, is a mixture of Poisson distributions. When mean and variance are fixed, it produces heavier right tails than Negative Binomial, so the GPD has practical advantages when modeling empirical counting distributions with heavy tails. A practical disadvantage of the GPD is that unlike the Negative Binomial distribution, it is not a member of the exponential family of distributions. This talk resulted from my desire to understand the GDP, which came to my attention in an applied setting. This talk is therefore exposition, not research, and it exemplifies how an applied topic can evoke connections to several standard but pleasant theoretical topics.

In essence, it explores the mathematical garden giving rise to the GPD. The most direct interpretation of the GPD is that it gives the distribution of the total family size in a class of branching process. The Lagrange expansion applied to a combinatorial generating function proves that the total family size of each branching process follows a GPD. The functional equation underlying the Lagrange expansion specifies an analytic function in one complex variable, and analytic continuation of the combinatorial generation function proves that the GPD is indeed a mixture of Poisson distributions. (The proof is not constructive, and the mixture distribution appears not to have a closed form.) Furthermore, through a standard correspondence, the branching process maps to the customer count in the initial busy period of a queueing process. The count distribution in the busy period gives another interpretation for the GPD. In the queueing interpretation, one particularly annoying factor in the GPD requires the Bertrand Ballot Theorem, evoking another standard method in queueing theory.