A zero-and-one inflated Poisson model and its application

To model count data with excess zeros and excess ones, Melkersson and Olsson (1999) proposed a zero-and-oneinflated Poisson (ZOIP) distribution. Zhang, Tian and Ng (2016) studied the properties and likelihood-based inference methods on ZOIP model. However, they only propose some estimation methods for the ZOIP model. In this paper, the maximum likelihood estimation (MLE) and Bayesian estimation for this model are investigated and some properties are derived. The reference prior and the Jeffreys prior are derived for this model. It is further shown that they are secondorder matching priors and the posterior distributions based on these priors are proper under a relatively mild condition. And the zero-and-one-inflated Poisson regression model has also been discussed. A simulation study based on proposed sampling algorithm is conducted to assess the performance of the proposed estimation for various sample sizes. Finally, two real data sets are analyzed to illustrate the practicability of the proposed method.