TY - JOUR
T1 - On the computation of multivariate scenario sets for the skew-t and generalized hyperbolic families
AU - Giorgi, Emanuele
AU - McNeil, Alexander J.
PY - 2014/2/4
Y1 - 2014/2/4
N2 - 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.
AB - 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.
KW - Angular symmetry
KW - Expectile depth
KW - Generalized hyperbolic distribution
KW - Half-space depth
KW - Multivariate scenario sets
KW - Skew-t distribution
UR - http://www.scopus.com/inward/record.url?scp=84904084488&partnerID=8YFLogxK
U2 - 10.1016/j.csda.2014.06.024
DO - 10.1016/j.csda.2014.06.024
M3 - Article
AN - SCOPUS:84904084488
SN - 0167-9473
VL - 100
SP - 205
EP - 220
JO - Computational Statistics & Data Analysis
JF - Computational Statistics & Data Analysis
ER -