Polya-Aeppli Noncentral Chi-Square Process and its Applications in Risk Analysis

Abstract

In this paper we introduce Polya-Aeppli non-central chi-square process as a mixed Polya-Aeppli process with mixing random variable having non-central chi-square distribution. We derive expression for PMF and discuss several properties. We consider a risk model with Polya-Aeppli non-central chi- square process as the counting process. The joint distribution of the time to ruin and deficit at the time of ruin is derived. The differential equation of the ruin probability is given. As example, we consider the case of exponentially distributed claims. Introduction:- Polya-Aeppli process was introduced by Minkova(2004) as a compound Poisson process with geometric compounding distribution. It is a generalization of homogeneous Poisson process and is used to model over-dispersed count data. In order to allow a for lack of homogeneity, some random variation is introduced in the parameter  (see Minkova(2013))of Polya-Aeppli process. This leads to the notion of a mixed Polya-Aeppli process. It is a modification of Polya-Aeppli process. One reason of the interest on these mixed Polya-Aeppli process lies on the fact that they are over-dispersed relative to Polya-Aeppli process and offer more flexibility than Polya-Aeppli process. Recently, many researchers used mixed Polya- Aeppli process as a claim counting process in risk modeling. I-Polya process was introduced by Minkova (2011) as a mixed Polya-Aeppli process with gamma mixing distribution. It is a generalization of the classical Polya process. Lazarova and Minkova(2015) studied Polya-Aeppli process with shifted gamma mixing distribution and called it I-Delaporate process. If 0 =  , I-Delaporate process reduces to Delaporate process. In this study we, introduce a new mixed Polya-Aeppli distribution which is called the Polya-Aeppli non-central chi-square distribution. It is a mixture of Polya-Aeppli distribution by mixing the Polya-Aeppli distribution and non-central chi-square distribution. Then we define a counting process with Polya-Aeppli non-central chi-square distribution and consider the risk model with Polya-Aeppli non-central chi-square counting process . The motivation behind to make a choice of non-central chi-square distribution as the mixing distribution is that it can view as a Poisson mixture of certain gamma distribution and it has various financial applications.