Analytical expressions of risk quantities for composite models

José María Sarabia and Enrique Calderín-Ojeda

Abstract

Composite models have received a lot of attention in the recent actuarial literature. In this paper, we obtain analytic expressions of different actuarial and statistical quantities for a general class of composite models derived from the McDonald’s family of probability distributions. These quantities include probability density functions, cumulative distribution functions, quantile functions, raw moments, value-at-risk, tail moments, moments of loss variables and the quantile density function, which is very useful for the computation of expectations of order statistics. Finally, applications of these models to two sets of unimodal and positively skewed insurance claim size data are provided.