Abstract
Let \(X_{1},\ldots,X_{n}\) be unit gamma Gompertz \((UGG)\) random variables with \(X_{i}\sim UGG(\alpha_{i},\beta_{i},\mu_{i};G)\) for \(i=1,\ldots,n\) and \(I_{p_{1}},\ldots,I_{p_{n}}\) are independent Bernoulli random variables, independent of the \(X_{i}\) ’s, with \(E(I_{p_{i}})=p_{i}\) , \(i=1,\ldots,n\) . Let \(Y_{i}=I_{p_{i}}X_{i}\) , for \(i=1,\ldots,n\) . In actuarial science, \(Y_{i}\) corresponds to the claim amount in a portfolio of risks. In this paper, we establish usual stochastic order and reversed hazard rate order between the largest claim amounts, by using the concept of vector majorization and related orders, when claim severities are independent. We also discuss stochastic comparisons between the smallest claim amounts in the sense of the usual stochastic order when claim severities are dependent. Further, we apply the results for some special cases of the unit gamma Gompertz model with possibly different parameters to illustrate.
