Stochastic comparisons of the smallest and largest claim amounts with location-scale claim severities

Suppose X 1 , … , X n are independent location-scale ( L S ) random variables with X i ∼ L S ( λ i , μ i ) , for i = 1 , … , n , and I p 1 , … , I p n are independent Bernoulli random variables, independent of the X i ’s, with E ( I p i ) = p i , i = 1 , … , n . Let Y i = I p i X i , for i = 1 , … , n . In actuarial science, Y i corresponds to the claim amount in a portfolio of risks. In this paper, we establish usual stochastic order between …