Some intriguing properties of Tukey's half-space depth

TL;DR

This paper explores the properties of Tukey's half-space depth and median in multivariate and Banach space contexts, revealing both intuitive and counterintuitive behaviors with significant implications for multivariate analysis.

Contribution

It derives new properties of Tukey's half-space depth and median, including extensions to infinite-dimensional spaces, highlighting their complex behaviors and statistical consequences.

Findings

Counterintuitive properties of half-space depth and median in multivariate data

Extension of half-space depth to Banach spaces with anomalous behaviors

Implications for multivariate analysis and infinite-dimensional data

Abstract

For multivariate data, Tukey's half-space depth is one of the most popular depth functions available in the literature. It is conceptually simple and satisfies several desirable properties of depth functions. The Tukey median, the multivariate median associated with the half-space depth, is also a well-known measure of center for multivariate data with several interesting properties. In this article, we derive and investigate some interesting properties of half-space depth and its associated multivariate median. These properties, some of which are counterintuitive, have important statistical consequences in multivariate analysis. We also investigate a natural extension of Tukey's half-space depth and the related median for probability distributions on any Banach space (which may be finite- or infinite-dimensional) and prove some results that demonstrate anomalous behavior of half-space depth in infinite-dimensional spaces.