# An Alternative Distribution for Modelling Overdispersion Count Data: Poisson Shanker Distribution

## Authors

• A Meytrianti Department of Mathematics, Universitas Indonesia, Depok, 16424, Indonesia
• S Nurrohmah Department of Mathematics, Universitas Indonesia, Depok, 16424, Indonesia
• M Novita Department of Mathematics, Universitas Indonesia, Depok, 16424, Indonesia

## Keywords:

maximum likelihood estimation, mixing, overdispersion, Poisson distribution, Shanker distribution

## Abstract

Poisson distribution is a common distribution for modelling count data with assumption mean and variance has the same value (equidispersion). In fact, most of the count data have mean that is smaller than variance (overdispersion) and Poisson distribution cannot be used for modelling this kind of data. Thus, several alternative distributions have been introduced to solve this problem. One of them is Shanker distribution that only has one parameter. Since Shanker distribution is continuous distribution, it cannot be used for modelling count data. Therefore, a new distribution is offered that is Poisson-Shanker distribution. Poisson-Shanker distribution is obtained by mixing Poisson and Shanker distribution, with Shanker distribution as the mixing distribution. The result is a mixture distribution that has one parameter and can be used for modelling overdispersion count data. In this paper, we obtain that Poisson-Shanker distribution has several properties are unimodal, overdispersion, increasing hazard rate, and right skew. The first four raw moments and central moments have been obtained. Maximum likelihood is a method that is used to estimate the parameter, and the solution can be done using numerical iterations. A real data set is used to illustrate the proposed distribution. The characteristics of the Poisson-Shanker distribution parameter is also obtained by numerical simulation with several variations in parameter values and sample size. The result is MSE and bias of the estimated parameter theta will increase when the parameter value rises for a value of n and will decrease when the value of n rises for a parameter value.