Bizarre mixed Poisson distributions and wastewater-based epidemiology
Abstract:
Mixed Poisson families are widely used to model count data with overdispersion, zero inflation, or heavy tails in a variety of applications including finance, biology, and the physical sciences. The Poisson rate is typically assigned a nonnegative-valued mixing distribution. Surprisingly, it is also possible for the mixing distribution to have negative support. For example, the Hermite distribution is analogous to mixing a Poisson with a Gaussian and can be derived using generating functions so long as constraints on the natural parameter are satisfied. I will discuss general conditions on the mixing distribution that are necessary for a mixed Poisson to exist. A key tool is the use of subweibull bounds on the rates of tail decay and Lp norm growth. I will illustrate the scope of mixed Poisson families with examples having different tail behaviors and comment on the mixed Poisson analog of the skewed extreme stable family. Time permitting, I will also briefly present some recent applied work in wastewater-based epidemiology.