On the Bivariate Composite Gumbel–Pareto Distribution

Abstract

In this paper, we propose a bivariate extension of univariate composite (two-spliced) distributions defined by a bivariate Pareto distribution for values larger than some thresholds and by a bivariate Gumbel distribution on the complementary domain. The purpose of this distribution is to capture the behavior of bivariate data consisting of mainly small and medium values but also of some extreme values. Some properties of the proposed distribution are presented. Further, two estimation procedures are discussed and illustrated on simulated data and on a real data set consisting of a bivariate sample of claims from an auto insurance portfolio. In addition, the risk of loss in this insurance portfolio is estimated by Monte Carlo simulation.

Keywords

• bivariate composite (two-spliced) distribution
• Gumbel’s bivariate exponential distribution
• bivariate Pareto of the first kind distribution
• maximum likelihood estimation procedure
• risk of loss