Mixed Poisson families with real-valued mixing distributions

TL;DR

This paper demonstrates that mixed Poisson distributions can be extended to include real-valued mixing distributions, broadening their applicability to model complex count data with overdispersion and heavy tails.

Contribution

It shows that the traditional nonnegative support assumption for mixing distributions is unnecessary, allowing for real-valued mixing distributions with specific tail properties.

Findings

Real-valued mixing distributions are possible for mixed Poisson models.

Examples include the mixed Poisson-extreme stable family with power law tails.

The mixing distribution needs a light left tail and some negative mass.

Abstract

Mixed Poisson distributions provide a flexible approach to the analysis of count data with overdispersion, zero inflation, or heavy tails. Since the Poisson mean must be nonnegative, the mixing distribution is typically assumed to have nonnegative support. We show this assumption is unnecessary and real-valued mixing distributions are also possible. Informally, the mixing distribution merely needs to have a light (subexponential) left tail and a small amount of probability mass on negative values. We provide several concrete examples, including the mixed Poisson-extreme stable family, where the mixing distribution has a power law tail.