Allocations of policy limits and ordering relations for aggregate remaining claims

Let X1,…,Xn be a set of n risks, with decreasing joint density function f, faced by a policyholder who is insured for this n risks, with upper limit coverage for each risk. Let l=(l1,…ln) and l∗=(l1∗,…ln∗) be two vectors of policy limits such that l∗ is majorized by l. It is shown that ∑i=1n(Xi−li)+ is larger than ∑i=1n(Xi−li∗)+ according to stochastic dominance if f is exchangeable. It is also shown that ∑i=1n(Xi−l(i))+ is larger than ∑i=1n(Xi−l(i)∗)+ according to stochastic dominance if either f is a decreasing arrangement or X1,…,Xn are independent and ordered according to the reversed hazard rate ordering. We apply the new results to multivariate Pareto distribution.