On the convex transform and right-spread orders of smallest claim amounts

Suppose Xλ1,…,Xλn is a set of Weibull random variables with shape parameter α>0, scale parameter λi>0 for i=1,…,n and Ip1,…,Ipn are independent Bernoulli random variables, independent of the Xλi's, with E(Ipi)=pi, i=1,…,n. Let Yi=XλiIpi, for i=1,…,n. In particular, in actuarial science, it corresponds to the claim amount in a portfolio of risks. In this paper, under certain conditions, we discuss stochastic comparison between the smallest claim amounts in the sense of the right-spread order. Moreover, while comparing these two smallest claim amounts, we show that the right-spread order and the increasing convex orders are equivalent. Finally, we obtain the results concerning the convex transform order between the smallest claim amounts and find a lower and upper bound for the coefficient of variation. The results established here extend some well-known results in the literature.