Extreme Geometric Quantiles Under Minimal Assumptions, with a Connection to Tukey Depth

TL;DR

This paper investigates the extremal properties of geometric quantiles, establishing bounds without moment assumptions and revealing a novel link to Tukey depth and distribution characterization.

Contribution

It introduces new extremal bounds for geometric quantiles that are independent of moment conditions and connects these bounds to Tukey depth, enhancing understanding of multivariate distribution behavior.

Findings

Established lower and upper bounds for extremal geometric quantiles.

Linked bounds to univariate quantiles and Tukey depth regions.

Provided insights into distribution behavior through geometric quantile analysis.

Abstract

Geometric (also known as spatial) quantiles, introduced by Chaudhury and representing one of the three principal approaches to defining multivariate quantiles, have been well studied in the literature. In this work, we focus on the extremal behaviour of these quantiles. We establish new extremal properties, namely general lower and upper bounds for the norm of extreme geometric quantiles, free of any moment conditions. We discuss the impact of such results on the characterization of distribution behaviour. Importantly, the lower bound can be directly linked to univariate quantiles and to halfspace (Tukey) depth central regions, highlighting a novel connection between these two fundamental notions of multivariate quantiles.