Halfspace depth does not characterize probability distributions

TL;DR

This paper demonstrates that the halfspace depth, a statistical measure used to characterize probability distributions, cannot uniquely identify all distributions as different distributions can have identical depth functions.

Contribution

It provides counterexamples showing that halfspace depth does not uniquely characterize multivariate probability distributions, challenging previous assumptions.

Findings

Multiple distinct distributions share the same halfspace depth everywhere.

Halfspace depth is not a complete descriptor of probability distributions.

Counterexamples highlight limitations of halfspace depth in distribution characterization.

Abstract

We give examples of different multivariate probability distributions whose halfspace depths coincide at all points of the sample space.