On a Bivariate Distribution with Composite Exponential-Pareto Marginals and Dependence in Low-Cost Claims

Abstract

Spliced distributions with Pareto tail, better known as compound distributions in the actuarial field, have been intensively studied in the univariate case; however, extending spliced distributions to the bivariate setting is an open problem, so that applications in the bivariate case are rather scarce. In this article, we present an application with auto insurance claim costs that illustrates the importance of using these bivariate distributions in quantifying the risk of loss. We define a bivariate Farlie–Gumbel–Morgenstern (FGM) distribution with Exponential–Pareto marginals, which are considered with and without the continuity condition at the threshold where the exponential changes to Pareto. After presenting some properties of this bivariate distribution, we discuss a proposed estimation procedure, which is not obvious since the marginal thresholds are unknown parameters and they have to be estimated separately. The properties of the estimation procedure are analyzed through a simulation study. We obtain that the mean square error (MSE) and the mean absolute error (MAE) decrease when the sample size increases for a set of positive dependence structures. We also use a real data sample of bivariate claims costs collected from an auto insurance portfolio in Spain. The results show how, when the dependence is taken into account, the shape of the total loss distribution changes affecting the risk of loss.