Stochastic comparisons of the smallest and largest claim amounts with location-scale claim severities

Abstract

Suppose X1,…,Xn are independent location-scale (LS) random variables with Xi∼LS(λi,μi), for i=1,…,n, and Ip1,…,Ipn are independent Bernoulli random variables, independent of the Xi’s, with E(Ipi)=pi, i=1,…,n. Let Yi=IpiXi, for i=1,…,n. In actuarial science, Yi corresponds to the claim amount in a portfolio of risks. In this paper, we establish usual stochastic order between the largest claim amounts when the matrix of parameters changes to another matrix in a certain mathematical sense. Under certain conditions, by using the concept of vector majorization and related orders, we also discuss stochastic comparisons between the smallest claim amounts in the sense of the usual stochastic and hazard rate orders in two cases: (i) when claim severities are dependent, and then (ii) when claim severities are independent. We then apply the results for some special cases of the location-scale model with possibly different scale and location parameters to illustrate the established results.