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To model count data with excess zeros and excess ones, in their unpublished manuscript, Melkersson and Olsson (1999) extended the zero-inflated Poisson distribution to a zero-and-one-inflated Poisson (ZOIP) distribution. However, the distributional theory and corresponding properties of the ZOIP have not yet been explored, and likelihoodbased inference methods for parameters of interest were not well developed. In this paper, we extensively study the ZOIP distribution by first constructing five equivalent stochastic representations for the ZOIP random variable and then deriving other important distributional properties. Maximum likelihood estimates of parameters are obtained by both the Fisher scoring and expectation—maximization algorithms. Bootstrap confidence intervals for parameters of interest and testing hypotheses under large sample sizes are provided. Simulations studies are performed and five real data sets are used to illustrate the proposed methods.

Properties of the zero-and-one inflated Poisson distribution and likelihood-based inference methods