Center-outward quantiles and the measurement of multivariate risk

TL;DR

This paper introduces a novel approach to multivariate risk measurement using center-outward quantiles based on measure transportation, providing new empirical risk measures and applications to regularly varying distributions.

Contribution

It develops smooth approximations for center-outward quantiles, enabling computation of new empirical risk measures and advancing multivariate risk analysis methods.

Findings

New empirical risk measures based on convex potentials

Effective approximation methods for center-outward quantiles

Applications demonstrate improved risk assessment in multivariate cases

Abstract

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. We propose a class of smooth approximations 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.