Stochastic comparisons of largest claim amounts from heterogeneous portfolios

Abstract

This paper investigates stochastic comparisons of largest claim amounts of two sets of independent or interdependent portfolios in the sense of some stochastic orders. Let random variable Xi$$ {X}_i $$ ( i=1,…,n$$ i=1,\dots, n $$ ) with distribution function F(x;αi)$$ F\left(x;{\alpha}_i\right) $$ , represents the claim amount for ith risk of a portfolio. Here two largest claim amounts are compared considering that the claim variables follow a general semiparametric family of distributions having the property that the survival function F‾(x;α)$$ \overline{F}\left(x;\alpha \right) $$ is increasing in α$$ \alpha $$ or is increasing and convex/concave in α$$ \alpha $$ . The results obtained in this paper apply to a large class of well‐known distributions including the family of exponentiated/generalized distributions (e.g., exponentiated exponential, Weibull, gamma and Pareto family), Rayleigh distribution and Marshall–Olkin family of distributions. As a direct consequence of some main theorems, we also obtained the results for scale family of distributions. Several numerical examples are provided to illustrate the results.