Ordering properties of the smallest and largest claim amounts in a general scale model

Suppose Xλ1,…,Xλn$ X_{\lambda _1},\ldots ,X_{\lambda _n} $ is a set of non-negative random variables with Xλi$ X_{\lambda _i} $ having the distribution function G(λit)$ G(\lambda _i\, t) $, λi>0$ \lambda _i>0 $ for i=1,…,n,$ i=1,\ldots ,n, $ and Ip1,…,Ipn$ I_{p_1},\ldots ,I_{p_n} $ are independent Bernoulli random variables, independent of the Xλi$ X_{\lambda _i} $’s, with E(Ipi)=pi$ E(I_{p_i})=p_i $, i=1,…,n$ i=1,\ldots ,n $. Let Yi=IpiXλi$ Y_{i}=I_{p_i} X_{\lambda _i} $, for i=1,…,n$ i=1, \ldots , n $. It is of interest to note that in actuarial science, Yi$ Y_{i} $ corresponds to the claim amount in a portfolio of risks. In this paper, under certain conditions, by using the concept of vector majorization and related orders, we discuss stochastic comparison between the smallest claim amount in the sense of the usual stochastic and hazard rate orders. We also obtain the usual stochastic order between the largest claim amounts when the matrix of parameters (h(p),λ)$ ({\boldsymbol{h(p)}}, {\boldsymbol{\lambda }}) $ changes to another matrix in a mathematical sense. We then apply the results for three special cases of the scale model: generalized gamma, Marshall–Olkin extended exponential and exponentiated Weibull distributions with possibly different scale parameters to illustrate the established results.