Statistics and Its Interface

Volume 11 (2018)

Number 2

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

Pages: 339 – 351



Wenchen Liu (School of Statistics, East China Normal University, Shanghai, China)

Yincai Tang (School of Statistics, East China Normal University, Shanghai, China)

Ancha Xu (College of Mathematics and Information Science, Wenzhou University, Zhejiang, China)


To model count data with excess zeros and excess ones, Melkersson and Olsson (1999) proposed a zero-and-one-inflated 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 second-order 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.


zero-and-one-inflated Poisson model, objective Bayes, reference prior, Metropolis-Hastings algorithm

Yincai Tang and Ahcna Xu were supported by the NSF of China (11271136, 81530086, 11671303, 11201345), by The 111 Project (B14019), by the NSF of Zhejiang Province (LY15G010006), and by the CPSF (2015M572598).

Received 10 April 2016

Published 7 March 2018