Halfspace depth for general measures: The ray basis theorem and its consequences

TL;DR

This paper explores the theoretical foundations of halfspace depth in multivariate analysis, providing a new elementary proof of the ray basis theorem, and discusses its implications for robust estimation and computation.

Contribution

It offers a novel, minimal-assumption proof of the ray basis theorem and links depth-trimmed regions to convex geometric concepts, enhancing understanding of depth properties.

Findings

Elementary proof of the ray basis theorem under minimal assumptions

Connections established between depth regions, floating bodies, and convex sets

Conditions for strict monotonicity of the halfspace depth

Abstract

The halfspace depth is a prominent tool of nonparametric multivariate analysis. The upper level sets of the depth, termed the trimmed regions of a measure, serve as a natural generalization of the quantiles and inter-quantile regions to higher-dimensional spaces. The smallest non-empty trimmed region, coined the halfspace median of a measure, generalizes the median. We focus on the (inverse) ray basis theorem for the halfspace depth, a crucial theoretical result that characterizes the halfspace median by a covering property. First, a novel elementary proof of that statement is provided, under minimal assumptions on the underlying measure. The proof applies not only to the median, but also to other trimmed regions. Motivated by the technical development of the amended ray basis theorem, we specify connections between the trimmed regions, floating bodies, and additional equi-affine convex sets related to the depth. As a consequence, minimal conditions for the strict monotonicity of the depth are obtained. Applications to the computation of the depth and robust estimation are outlined.