Ordering the smallest claim amounts from two sets of interdependent heterogeneous portfolios
Let X_λ_1,...,X_λ_n be a set of dependent and non-negative random variables share a survival copula and let Y_i= I_p_iX_λ_i, i=1,...,n, where I_p_1,...,I_p_n be independent Bernoulli random variables independent of X_λ_i's, with E[I_p_i]=p_i, i=1,...,n. In actuarial sciences, Y_i corresponds to the claim amount in a portfolio of risks. This paper considers comparing the smallest claim amounts from two sets of interdependent portfolios, in the sense of usual and likelihood ratio orders, when the variables in one set have the parameters λ_1,...,λ_n and p_1,...,p_n and the variables in the other set have the parameters λ^*_1,...,λ^*_n and p^*_1,...,p^*_n. Also, we present some bounds for survival function of the smallest claim amount in a portfolio. To illustrate validity of the results, we serve some applicable models.
READ FULL TEXT