Bounding quantiles of Wasserstein distance between true and empirical measure

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.