The Touchard distribution

Raul Matsushita, Donald Pianto, Bernardo B. De Andrade, Andre Cançado, Sergio Da Silva

Communications in Statistics-Theory and Methods 48 (2019) 2049-2059

https://doi.org/10.1080/03610926.2018.1444177

We present a novel model, which is a two-parameter extension of the Poisson distribution. Its normalizing constant is related to the Touchard polynomials, hence the name of this model. It is a flexible distribution that can account for both under- or overdispersion and concentration of zeros that are frequently found in non-Poisson count data. In contrast to some other generalizations, the Hessian matrix for maximum likelihood estimation of the Touchard parameters has a simple form. We exemplify with three data sets, showing that our suggested model is a competitive candidate for fitting non-Poisson counts.