On stochastic comparison between aggregate claim amounts

Let Xλ1,…,XλnXλ1,…,Xλn be nonnegative independent random variables with XλiXλi having survival function F¯(.,λi), i=1,…,ni=1,…,n, where λi>0λi>0. Let Ip1,…,IpnIp1,…,Ipn be independent Bernoulli random variables independent of XλiXλi with E(Ipi)=piE(Ipi)=pi , i=1,…,ni=1,…,n. Further, assume that F¯(.,λi) is a decreasing and convex function with respect to λiλi, i=1,…,ni=1,…,n and that the survival function of ∑i=1nXλi is Schur-convex in λ=(λ1,…,λn)λ=(λ1,…,λn). In this paper we show that under the above settings the survival function of S(λ,p)=∑i=1nIpiXλi is Schur-convex in (λ1,g(p1)),…,(λn,g(pn))(λ1,g(p1)),…,(λn,g(pn)) with respect to multivariate chain majorization, where g(p)=-logpg(p)=-logp or g(p)=(1-p)/pg(p)=(1-p)/p and p=(p1,…,pn)p=(p1,…,pn). We show an application of the main result in the case that the variables XλiXλi, i=1,…,ni=1,…,n, have Weibull or gamma distributions.