The zero-and-plus/minus-one inflated extended-Poisson distribution

In this paper, we introduce a new distribution defined on $$ , called the $$ distribution, which can be viewed as a natural extension of the zero-and-one-inflated Poisson ( $$ ) distribution. It is designed to fit the count data with potentially excess zeros and/or ones, and/or minus ones. We explore its various properties and investigate the estimation of the unknown parameters. Moreover, simulation experiments are carried out to attest to the performance of the estimation. Through the use of a useful data set on football scores, the applicability of the proposed distribution is examined.