Center-outward quantiles and the measurement of multivariate risk

All multivariate extensions of the univariate theory of risk measurement run into the same fundamental problem of the absence, in dimension d>1, of a canonical ordering of Rd. Based on measure transportation ideas, several attempts have been made recently in the statistical literature to overcome that conceptual difficulty. In Hallin (2017), the concepts of center-outward distribution and quantile functions are developed as generalizations of the classical univariate concepts of distribution and quantile functions, along with their empirical versions. The center-outward distribution function F± is a cyclically monotone mapping from Rd to the open unit ball Bd, while its empirical counterpart F±(n) is obtained as a cyclically monotone mapping from the sample to a regular grid over Bd; in dimension d=1, F± reduces to 2F−1. Based on the concept of Moreau envelope, a smooth interpolation F˜± of F± has been proposed in del Barrio et al. (2018). Here, we suggest to adapt the definition of the empirical F±(n) so as to relax the presence of ties, which is impractical in the context of risk measurement and propose a class of smooth approximations Fn,ξ (ξ a smoothness index) of F±(n) as an alternative to the F˜± interpolation. Associated with the concepts of center-outward distribution and quantile functions and the associated convex potentials, we construct measures of risk of the maximum correlation type and their estimators based on F±(n) and Fn,ξ. We also discuss the use of the volumes of the resulting empirical quantile regions. Some simulations and applications to case studies illustrate the value of the approach.