The long quest for quantiles and ranks in ${\mathbb R}^d$ and on manifolds

Abstract

Quantiles are a fundamental concept in probability, and an essential tool in statistics, from descriptive to inferential. Still, despite half a century of attempts, no satisfactory and fully agreed-upon definition of the concept, and the ``dual'' notion of ranks, is available beyond the well-understood case of univariate variables and distributions. The need for such a definition is particularly critical for variables taking values in ${\mathbb R}^d$, for directional variables (values on the hypersphere), and, more generally, for variables with values on manifolds. Unlike the real line, indeed, no canonical ordering is available on these domains. We show how measure transportation brings a solution to this problem by characterizing distribution-specific (data-driven, in the empirical case) orderings and {\it center-outward} distribution and quantile functions (ranks and signs in the empirical case) that satisfy all the properties expected from such concepts while reducing, in the case of real-valued variables, to the classical univariate notion.