Comparison of multivariate risks and positive dependence

In this paper we extend some recent results on the comparison of multivariate risk vectors w.r.t. supermodular and related orderings. We introduce a dependence notion called ‘weakly conditional increasing in sequence order’ that allows to conclude that ‘more dependent’ vectors in this ordering are also comparable w.r.t. the supermodular ordering. At the same time this ordering allows to compare two risks w.r.t. the directionally convex order if the marginals increase convexly. We further state comparison criteria w.r.t. the directionally convex order for some classes of risk vectors which are modelled by functional influence factors. Finally we discuss Fréchet-bounds w.r.t. ∆-monotone functions when multivariate marginals are given. It turns out that comonotone vectors in the case of multivariate marginals no longer yield necessarily the largest risks but even may in some cases be vectors which minimize risk.