Criteria for the Stochastic Ordering of Random Sums, with Actuarial Applications

It is shown that vectors ( S M 1 , … , S Mn ) and ( S' M'1 , …, S' M'n ) of random sums of positive random variables are stochastically ordered by upper orthant dependence, lower orthant dependence, concordance or by the supermodular ordering whenever their corresponding random numbers of terms ( M 1 , … , M n ) and ( M' 1 , … , M' n ) are themselves ordered in this fashion. Actuarial applications of these results are given to different dependence structures for the collective risk model with several classes of business.

Keywords:

• Adjustment Coefficient
• Collective Risk Model
• Comonotonicity
• Concordance
• Copulas
• Lower Orthant Dependence
• Positive Quadrant Dependence
• Upper Orthant Dependence
• Random Sums
• Supermodular Ordering