Inference concerning the size of the zero class from an incomplete poisson sample

Abstract

In this paper, we consider a model in which the units associated with some population are assumed to be at risk with respect to some mechanism. Units succumbing to this risk mechanism shall be said to be “infected“, and are assumed to generate independent and identically distributed Poisson counts. Only positive counts are observed. The analysis of this model is complicated by the fact that a unit may generate no observable counts either because it was not infected, or because the number of counts actually generated has fallen into the zero class of the Poisson distribution. Inference is sought with respect to the number of infected units generating no observed counts, the probability of infection, and the parameter specifying the mean of the Poisson distribution. A Bayesian analysis is considered so that prior information regarding the probability of infection may be incorporated. Both an independent and a dependent mechanism for infection will be considered. The dependent case will involve a quasi-binomial distribution. An example relating to an epidemic of cholera in a village in India motivates the model and our analysis.