Open Access Journal of Biostatistics and Biometrics

The Discrete Poisson-Shanker Distribution

*Rama Shanker
Department Of Biostatistics, Eritrea Institute Of Technology, Asmara,, Eritrea

*Corresponding Author:
Rama Shanker
Department Of Biostatistics, Eritrea Institute Of Technology, Asmara,, Eritrea
Email:shankerrama2009@gmail.co

Published on: 2018-07-27

Abstract

In this paper, Poisson-Shanker distribution (PSD) has been obtained by compounding Poisson distribution with Shanker distribution introduced by Shanker [1]. The first four moments about origin and the moments about mean have been obtained. The expressions for coefficients of variation, skewness and kurtosis have been obtained. Mathematical and statistical properties of the PSD including increasing hazard rate, unimodality, over-dispersion, and generating function have been discussed. The estimation of its parameter has been discussed using the method of maximum likelihood and the method of moments. The distribution has been fitted to some data sets to test its goodness of fit over Poisson distribution and Poisson-Lindley distribution (PLD) introduced by Sankaran [2].

Keywords

Shanker Distribution; Compounding; Moments; Statistical Properties; Estimation of Parameters; Goodness of Fit

Introduction

A number of discrete distributions have been introduced in Statistics and the main reason for so many discrete distributions is that each distribution is based on certain assumptions and has specific applications due to its shape and distributional properties. It has been observed that a particular distribution will not fit well on all discrete data due to various reasons including the variation in the data, shape of the distribution, assumptions of the distribution, some among others. Therefore, a search for new discrete distribution is still going on which can fit data well as compared to existing distributions. Shanker distribution, as shown by Shanker [1], is a better model than the Lindley and exponential distributions for modeling lifetime data from biomedical science and engineering, it is expected that the Poisson mixture of Shanker distribution named Poisson-Shanker distribution (PSD) will provide a better fit than the Poisson-Lindley distribution (PLD) [2,5], a Poisson mixture of Lindley distribution [3,4], for modeling count data. Further, since both PLD and PSD are of one parameter, a comparative study between PLD and PSD is justifiable from fitting discrete data. The main objective of the paper is to firstly find the Poisson mixture of Shanker distribution and have detailed study about its distributional properties, estimation of parameter and applications over some discrete distributions. In the present paper, a Poisson mixture of Shanker distribution introduced by Shanker [1] named, “Poisson-Shanker distribution (PSD)” has been proposed. Its various mathematical and statistical properties including its shape, moments, coefficient of variation, skewness, kurtosis, index of dispersion, hazard rate, over-dispersion, unimodality etc have been discussed. The estimation of its parameter has been discussed using maximum likelihood estimation and method of moments. The goodness of fit of PSD along with Poisson distribution and Poisson-Lindley distribution (PLD), a Poisson mixture of Lindley [3] distribution and introduced by Sankaran [2], have been discussed.