Likelihood Inference for Type I Bivariate Polya-Aeppli Distribution

Abstract

The Poisson distribution is commonly used in analyzing count data, and many insurance companies are interested in studying the related risk models and ruin probability theory. Over the past century, many different bivariate models have been developed in the literature. The bivariate Poisson distribution was first introduced by Campbell (1934) for modelling bivariate accident data. However, in some situations, a given dataset may possess over-dispersion compared to Poisson distribution which moti- vated researchers to develop alternative models to handle such situations. In this regard, Minkova and Balakrishnan (2014a) developed the Type I bivariate Polya- Aeppli distribution by using compounding with Geometric random variables and the trivariate reduction method. Inference for this Type I bivariate Polya-Aeppli distribution is the topic of this thesis. The parameters in a model are used to describe and summarize a given sample within a specific distribution. So, their estimation becomes important and the goal of estimation theory is to seek a method to find estimators for the parameters of interest that have some good properties. There exist many methods of finding estimators such as Method of Moments, Bayesian estimators, Least Squares, and Maximum Likelihood Estimators (MLEs). Each method of estimation has its own strength and weakness (Casella and Berger (2008)). Minkova and Balakrishnan (2014a) discussed the moment estimation of the parameters of the Type I bivariate Polya-Aeppli dis- tribution. In this thesis, we develop the likelihood inference for this model. A simulation study is carried out with various parameter settings. The obtained results show that the MLEs require more computational time compared to Moment estimation. However, Method of Moments (MoM) did not result in good estimates for all the simulation settings. In terms of mean squared error and bias, we observed that MLEs performed, in most of the settings, better than MoM. Finally, we apply the Type I bivariate Polya-Aeppli model to a real dataset containing the frequencies of railway accidents in two subsequent six year periods. We also carry out some hypothesis tests using the Wald test statistic. From these results, we conclude that the two variables belong to the same univariate Polya-Aeppli distribution but are correlated.