Multivariate Quantiles: Geometric and Measure-Transportation-Based Contours

TL;DR

The paper tackles the lack of a canonical multivariate quantile by contrasting two leading constructions: geometric quantiles and center-outward quantiles based on measure transport. It develops their theoretical properties, including regions and contours, monotonicity-style behavior, and regularity, and then performs a thorough empirical comparison through simulations that estimate contours via optimal couplings. The results show that center-outward quantile contours more faithfully capture the geometry of non-spherical distributions (e.g., banana-shaped, skew-t), while geometric contours can misrepresent extreme regions and even extend beyond the natural support. Overall, the work argues in favor of the measure-transport, center-outward framework as a robust, geometry-aware approach to multivariate quantiles with practical implications for data analysis and visualization.

Abstract

Quantiles are a fundamental concept in probability and theoretical statistics and a daily tool in their applications. While the univariate concept of quantiles is quite clear and well understood, its multivariate extension is more problematic. After half a century of continued efforts and many proposals, two concepts, essentially, are emerging: the so-called (relabeled) geometric quantiles, extending the characterization of univariate quantiles as minimizers of an L1 loss function involving the check functions, and the more recent center-outward quantiles based on measure transportation ideas. These two concepts yield distinct families of quantile regions and quantile contours. Our objective here is to present a comparison of their main theoretical properties and a numerical investigation of their differences.

Figures (2)

• Figure 1: Geometric (red) and measure-transportation-based (blue) contours of levels $\tau=.25, .5, .75$ for samples of size $N=2400$ drawn from four different distributions. Geometric contours are relabeled according to their probability content, and measure-transportation contours are obtained through the interpolation scheme (see Hallin et al. (2021)) of the optimal coupling between the observations and a $40\times60$ regular grid ${\mathfrak G}_N$ of the unit ball in ${\mathbb R}^2$; see Section 4.2 for details. Empirical quantiles, both geometric and center-outward, are connected by lines in the plots.
• Figure 2: Geometric (red) and measure-transportation-based (blue) contours of order $\tau=.90, .95, .99$ for samples of size $N=1,000$ drawn from a (nonspherical) bivariate Gaussian distribution on ${\mathbb R}^2$. Empirical quantiles, both geometric and center-outward, are connected by lines in the plots.