Concentration of the empirical level sets of Tukey's halfspace depth

TL;DR

This paper investigates how the upper level sets of Tukey's halfspace depth concentrate around their population counterparts, including a discretized version, providing insights into their stability and behavior in data analysis.

Contribution

It introduces new results on the concentration properties of Tukey's depth level sets and their discretized variants, enhancing understanding of their statistical stability.

Findings

Upper level sets of Tukey depth concentrate around their population versions

Discretized Tukey depth level sets also exhibit concentration properties

Results improve understanding of depth-based data analysis methods

Abstract

Tukey depth, aka halfspace depth, has attracted much interest in data analysis, because it is a natural way of measuring the notion of depth relative to a cloud of points or, more generally, to a probability measure. Given an i.i.d. sample, we investigate the concentration of upper level sets of the Tukey depth relative to that sample around their population version. We also study the concentration of the upper level sets of a discretized version of Tukey depth.