On higher-degree bivariate stop-loss transforms, with applications

Abstract

The bivariate extension of the higher degree stop-loss transforms is considered. The logarithmic partial derivatives of the higher degree bivariate stop-loss transforms define useful higher degree stoploss rate couples, which are related to the hazard rate and mean excess couples in the lower degrees. We show that the higher degree stop-loss rate couples determine the higher degree bivariate stop-loss transforms uniquely, and that the latter ones determine a bivariate survival function uniquely. As illustration, the set of all non-negative bivariate distributions with semi-linear affine mean excess couple is characterized. The higher degree bivariate stop-loss orders, which extend the well-known higher degree stop-loss orders to the bivariate case, are also considered. These orderings are weaker than the concordance or correlation order and define a hierarchy of orderings, which becomes weaker as the total degree increases. Concrete illustrations for bivariate diatomic risks, with an actuarial application to excess-of-loss on two lives, and for a class of bivariate risks with continuous margins are included.