On stochastic comparison between aggregate claim amounts

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Let X λ 1 , … , X λ n be nonnegative independent random variables with X λ i having survival function F ¯ ( ., λ i ) , i = 1 , … , n , where λ i > 0 . Let I p 1 , … , I p n be independent Bernoulli random variables independent of X λ i with E ( I p i ) = p i , i = 1 , … , n . Further, assume that F ¯ ( ., λ i ) is a decreasing and convex function with respect to λ i , i = 1 , … , n and that the survival function of ∑ i = 1 n X λ i is Schur-convex in λ = ( λ 1 , … , λ n ) . In this paper we show that under the above settings the survival function of S ( λ , p ) = ∑ i = 1 n I p i X λ i is Schur-convex in ( λ 1 , g ( p 1 ) ) , … , ( λ n , g ( p n ) ) with respect to multivariate chain majorization, where g ( p ) = - log p or g ( p ) = ( 1 - p ) / p and p = ( p 1 , … , p n ) . We show an application of the main result in the case that the variables X λ i , i = 1 , … , n , have Weibull or gamma distributions.