Partial reconstruction of measures from halfspace depth

TL;DR

This paper investigates how much information about a measure's support and mass distribution can be inferred from its halfspace depth function, despite the fact that the depth function does not uniquely determine the measure.

Contribution

It demonstrates that partial reconstruction of a measure's support and atomic parts is possible from its halfspace depth function, even though the measure is not uniquely determined.

Findings

Support and atomic parts of the measure can be partially reconstructed from halfspace depth.

Two different measures can share the same halfspace depth function.

A specific bivariate distribution's atomic part was successfully reconstructed.

Abstract

The halfspace depth of a $d$-dimensional point $x$ with respect to a finite (or probability) Borel measure $\mu$ in $\mathbb{R}^d$ is defined as the infimum of the $\mu$-masses of all closed halfspaces containing $x$. A natural question is whether the halfspace depth, as a function of $x \in \mathbb{R}^d$, determines the measure $\mu$ completely. In general, it turns out that this is not the case, and it is possible for two different measures to have the same halfspace depth function everywhere in $\mathbb{R}^d$. In this paper we show that despite this negative result, one can still obtain a substantial amount of information on the support and the location of the mass of $\mu$ from its halfspace depth. We illustrate our partial reconstruction procedure in an example of a non-trivial bivariate probability distribution whose atomic part is determined successfully from its halfspace depth.