On Heterogeneity in Wasserstein Space

TL;DR

This paper investigates heterogeneity among probability measures in Wasserstein space, proposing an estimator with strong statistical properties and methods to identify influential observations within a population.

Contribution

It introduces a novel estimator for heterogeneity in Wasserstein space that is unbiased, consistent, asymptotically normal, and stable under measure estimation.

Findings

Estimator is unbiased and consistent

Provides a method for comparing populations

Identifies observations contributing most to heterogeneity

Abstract

Data represented by probability measures arise as empirical distributions, posterior distributions, and feature-based representations of complex objects. We study heterogeneity in a population of probability measures through the expected value of a chosen transform of the pairwise Wasserstein distance. The resulting estimator is unbiased and, under simple moment conditions on the population law, is strongly consistent, asymptotically normal, and equipped with a consistent standard error. This also yields a simple comparison of two populations and remains stable under plug-in approximation when the measures are estimated. The associated empirical eccentricities identify the observations that contribute most strongly to heterogeneity within a sample.