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

Abstract

Let $X_{\lambda_1}, \ldots , X_{\lambda_n}$ be independent non-negative random variables belong to the transmuted-G model and let $Y_i=I_{p_i} X_{\lambda_i}$, $i=1,\ldots,n$, where $I_{p_1}, \ldots, I_{p_n}$ are independent Bernoulli random variables independent of $X_{\lambda_i}$'s, with ${\rm E}[I_{p_i}]=p_i$, $i=1,\ldots,n$. 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 $\lambda_1,\ldots,\lambda_n$ and the variables in the other set have the parameters $\lambda^{*}_1,\ldots,\lambda^{*}_n$. For illustration we apply the results to the transmuted-G exponential and the transmuted-G Weibull models.