On the breakdown point of transport-based quantiles

TL;DR

This paper investigates the robustness of transport-based multivariate quantiles, establishing their breakdown points and showing they share similar robustness properties with univariate quantiles.

Contribution

It provides the first analysis of the breakdown point for transport-based multivariate quantiles, linking their robustness to Tukey depth.

Findings

Transport median has a breakdown point of 1/2.

Transport depth contours have breakdown points equal to their order τ.

Multivariate transport depth shares robustness properties with univariate counterparts.

Abstract

Recent work has used optimal transport ideas to generalize the notion of (center-outward) quantiles to dimension $d\geq 2$. We study the robustness properties of these transport-based quantiles by deriving their breakdown point, roughly, the smallest amount of contamination required to make these quantiles take arbitrarily aberrant values. We prove that the transport median defined in Chernozhukov et al.~(2017) and Hallin et al.~(2021) has breakdown point of $1/2$. Moreover, a point in the transport depth contour of order $\tau\in [0,1/2]$ has breakdown point of $\tau$. This shows that the multivariate transport depth shares the same breakdown properties as its univariate counterpart. Our proof relies on a general argument connecting the breakdown point of transport maps evaluated at a point to the Tukey depth of that point in the reference measure.