Ordering Properties of the Smallest Claim Amounts from Two Heterogeneous Portfolios and Their Applications in Insurance

‎Suppose X_1,…,X_n is a set of non-negative random variables with X_i having‎ Exponentiated Weibull distribution for i=1,…,n‎ and I_1,…,I_n are independent Bernoulli random variables‎, ‎independent of the X_i s‎, ‎respectively‎, ‎with E(I_i )=p_i‎, i=1,…,n‎. ‎Let Y_i=I_i X_i, ‎for i=1,…,n. In applications‎, ‎actuarial science thus Y_i corresponds to claim amount in a portfolio of risks‎. ‎In this paper‎, ‎We obtain the usual stochastic order between the smallest claim amounts when the matrix of parameters (α,λ) changes to another matrix in a mathematical sense‎. ‎We also show that‎, ‎under some conditions on the common copula‎, the usual stochastic order of smallest claim amounts with heterogeneous claims is smaller than the smallest claim amounts with homogeneous‎ claims having a common survival function‎, ‎which is equal to the average of the survival functions‎ of the heterogeneous claims‎. ‎The results established here extend some well-known results in the literature‎.