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 generalisations of the classical univariate concepts of distribution and quantile functions, along with their empirical versions. The center-outward distribution function F± is a homeomorphic cyclically monotone mapping from Rd \ F−1 ± (0) to the open punctured unit ball Bd \ {0}, while its empirical counterpart F(n) ± is a cyclically monotone mapping from the sample to a regular grid over Bd. In dimension d = 1, F± reduces to 2F − 1, while F(n) ± generates the same sigma-field as traditional univariate ranks. The empirical F(n) ± ,however, involves a large number of ties, which is impractical in the context of risk measurement. We therefore propose a class of smooth approximations Fn,ξ (ξ a smoothness index) of F(n) ± as an alternative to the interpolation developed in del Barrio et al. (2018). This approximation allows for the computation of some new empirical risk measures, based either on the convex potential associated with the proposed transports, or on the volumes of the resulting empirical quantile regions. We also discuss the role of such transports in the evaluation of the risk associated with multivariate regularly varying distributions. Some simulations and applications to case studies illustrate the value of the approach.