Mixed Poisson families with real-valued mixing distributions

Oregon Convention Center 

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. Here we provide 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. We illustrate the scope of mixed Poisson families with examples having different tail behaviors. Finally we comment on the mixed Poisson analogs of the skewed stable and extreme stable Tweedie families.

probability distribution

Tweedie

count data

compound Poisson

exponential dispersion models 

Section on Bayesian Statistical Science