Bounding quantiles of Wasserstein distance between true and empirical measure

07/03/2019

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by   Samuel N. Cohen, et al.

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Consider the empirical measure, P̂_N, associated to N i.i.d. samples of a given probability distribution P on the unit interval. For fixed P the Wasserstein distance between P̂_N and P is a random variable on the sample space [0,1]^N. Our main result is that its normalised quantiles are asymptotically maximised when 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 P. We also suggest extensions to higher dimensions with numerical evidence.