On the computation of multivariate scenario sets for the skew-t and generalized hyperbolic families

The problem of computing multivariate scenarios sets for skewed distributions is motivated by the potential use of such sets in the stress testing of insurance companies and banks. Multivariate scenario sets based on the notion of half-space depth (HD) are considered and the notion of expectile depth (ED) is introduced. These depth concepts facilitate the definition of convex scenario sets, which generalize the concepts of quantiles and expectiles to higher dimensions. In the case of elliptical distributions the scenario sets coincide with the regions encompassed by the contours of the density function. In the context of multivariate skewed distributions, the equivalence of depth contours and density contours does not hold in general. Two parametric families that account for skewness and heavy tails are analysed: the generalized hyperbolic and the skew-t distributions. By making use of a canonical form representation, where skewness is completely absorbed by one component, it is shown that the HD contours of these distributions are near-elliptical; in the case of the skew-Cauchy distribution the HD contours are exactly elliptical. A measure of multivariate skewness as a deviation from angular symmetry is proposed. This measure is shown to explain the quality of the elliptical approximation for the HD contours.