Composite Exponential-Pareto distribution

Abstract

One of the few goals of statistical modeling is to see and analyze the probability of an event which can be represented with data. A probability distribution that is used for modeling data should have some abilities such as flexibility for modeling different kinds of data. Therefore, modeling data is of great importance. Furthermore, insurance companies also need to model data, which in this case is called modeling claim data. Modeling the claims distribution has its own challenge (e.g. skewed and heavy tailed) since most of the claim distributions are different from any classical distributions, therefore researchers are trying to find new models that can fit insurance data better. In this paper, a composite Exponential-Pareto distribution was proposed and introduced. This distribution is equal, but not equivalent to, an exponential density up to a certain threshold value, and a Pareto type-I density for the rest of the model. When being compared with the exponential distribution, the emerging density has a similar shape and a larger tail, and while being compared with the Pareto distribution, the emerging density has a smaller tail. A method to develop a composite distribution is called as composite parametric modeling, which introduced by Cooray and Ananda (2005). In this model, both the exponential distribution and the Pareto type-I distribution have the same weight. Based on the result, composite Exponential-Pareto distribution has some limitations, which are likely to severely diminish its potential for practical applications to real world insurance data. In order to address these issues, there are two different composite Exponential-Pareto distributions based on exponential and Pareto type-I distributions in order to address these concerns. These two different composite Exponential-Pareto distributions are based on the two-component mixture model introduced by Scollnik (2007). The first distribution, which is a reinterpreted composite Exponential-Pareto distribution from the first composite Exponential-Pareto distribution based on the two-component mixture model, has a fixed mixing weight. Meanwhile, the second distribution is a composite Exponential-Pareto distribution with a mixing weight that is not fixed so the distribution can be more flexible and can model different kinds of data. These three composite Exponential-Pareto distributions has k-th raw-moment that only defined for some k > 0. Therefore, this distribution can be categorized as a heavy-tail distribution. The result of this research is a composite distribution that could model a lot of data with characteristics such as unimodal, right-skewed, and heavy-tail because the composite distribution has similar characteristics. A data illustration was presented as a demonstration for how to implement the composite Exponential-Pareto distribution.