Multivariate Quantiles: Geometric and Measure-Transportation-Based Contours

TL;DR

This paper compares two emerging multivariate quantile concepts—geometric and measure-transportation-based—analyzing their theoretical properties and differences through numerical experiments.

Contribution

It provides a comprehensive comparison of geometric and measure-transportation-based multivariate quantiles, highlighting their theoretical distinctions and practical differences.

Findings

Geometric and measure-transportation quantiles form distinct families of regions.

Theoretical properties of the two quantile concepts differ significantly.

Numerical experiments illustrate practical differences in contour shapes and properties.

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.