Distortion risk measures for nonnegative multivariate risks

We apply distortion functions to bivariate survival functions for non-negative random variables. This leads to a natural extension of univariate distortion risk measures to the multivariate setting. For Gini's principle, the proportional hazard transform and the dual power transform distortions, certain families of multivariate distributions lead to a straightforward risk measure. We show that an exact analytical expression can be obtained in some cases. We consider the independence case, the bivariate Pareto distribution and the bivariate exponential distribution. An illustration of the estimation procedure and the interpretation is also included. In the case study we consider two loss events with one single risk value and monitor the two events together over four different periods. We conclude that the Dual Power Transform gives more weight to the observations of extreme losses, but that the distortion parameter can modulate this influence in all cases. In our example, multivariate risk clearly diminishes over time