# Bounding quantiles of Wasserstein distance between true and empirical measure

## TL;DR

This paper analyzes the asymptotic behavior of Wasserstein distance quantiles between true and empirical measures, identifying worst-case distributions and providing confidence regions, with extensions to higher dimensions.

## Contribution

It characterizes the asymptotic maximizers of Wasserstein quantiles and derives explicit confidence regions for the true distribution, extending to higher dimensions with numerical support.

## Key findings

- Asymptotic maximizers are convex combinations of two-point and uniform distributions.
- Explicit confidence regions for the true measure are derived.
- Numerical evidence supports extensions to higher dimensions.

## Abstract

Consider the empirical measure, $\hat{\mathbb{P}}_N$, associated to $N$ i.i.d. samples of a given probability distribution $\mathbb{P}$ on the unit interval. For fixed $\mathbb{P}$ the Wasserstein distance between $\hat{\mathbb{P}}_N$ and $\mathbb{P}$ is a random variable on the sample space $[0,1]^N$. Our main result is that its normalised quantiles are asymptotically maximised when $\mathbb{P}$ is a convex combination between the uniform distribution supported on the two points $\{0,1\}$ and the uniform distribution on the unit interval $[0,1]$. This allows us to obtain explicit asymptotic confidence regions for the underlying measure $\mathbb{P}$. We also suggest extensions to higher dimensions with numerical evidence.