On the Computation of Multivariate Scenario Sets for the Skew-t and Generalized Hyperbolic Families

Abstract

We examine the problem of computing multivariate scenarios sets for skewed distributions. Our interest is motivated by the potential use of such sets in the "stress testing" of insurance companies and banks whose solvency is dependent on changes in a set of financial "risk factors". We define multivariate scenario sets based on the notion of half-space depth (HD) and also introduce the notion of expectile depth (ED) where half-spaces are defined by expectiles rather than quantiles. We then use the HD and ED functions to define convex scenario sets that generalize the concepts of quantile and expectile to higher dimensions. In the case of elliptical distributions these 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. We consider two parametric families that account for skewness and heavy tails: the generalized hyperbolic and the skew-t distributions. By making use of a canonical form representation, where skewness is completely absorbed by one component, we show that the HD contours of these distributions are "near-elliptical" and, in the case of the skew-Cauchy distribution, we prove that the HD contours are exactly elliptical. We propose a measure of multivariate skewness as a deviation from angular symmetry and show that it can explain the quality of the elliptical approximation for the HD contours.