Bounding quantiles of Wasserstein distance between true and empirical measure
share
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.
READ FULL TEXT