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arXiv · 1907.02006

Bounding quantiles of Wasserstein distance between true and empirical measure

Samuel N. Cohen·Martin N. A. Tegnér·Johannes Wiesel
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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.

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Samuel N. Cohen, Martin N. A. Tegnér, Johannes Wiesel. 2019-07-03. Bounding quantiles of Wasserstein distance between true and empirical measure. https://arxiv.org/abs/1907.02006

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