Stochastic comparisons between the extreme claim amounts from two heterogeneous portfolios in the case of transmuted-G model

Abstract

Let $X_{{\lambda }_1},\dots ,X_{{\lambda }_n}$ be independent non-negative random variables belong to the transmuted-G model and let $Y_i=I_{p_i}{X}_{{\lambda }_i}$, $i=1,\dots ,n$, where $I_{p_1},\dots ,I_{p_n}$ are independent Bernoulli random variables independent of $X_{{\lambda }_i}$'s, with $\mathrm{E}[I_{p_i}]=p_i$, $i=1,\dots ,n$. In actuarial sciences, $Y_i$ corresponds to the claim amount in a portfolio of risks. In this paper we compare the smallest and the largest claim amounts of two sets of independent portfolios belonging to the transmuted-G model, in the sense of usual stochastic order, hazard rate order and dispersive order, when the variables in one set have the parameters ${\lambda }_1,\dots ,{\lambda }_n$ and the variables in the other set have the parameters ${\lambda }_1^{\ast },\dots ,{\lambda }_n^{\ast }$. For illustration we apply the results to the transmuted-G exponential and the transmuted-G Weibull models.