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The subject of this paper is the estimation of a probability measure on${\mathbb R}^d$ from data observed with an additive noise, under theWasserstein metric of order $p$ (with $p\geq 1$). We assume that thedistribution of the errors is known and belongs to a class of supersmoothdistributions, and we give optimal rates of convergence for the Wassersteinmetric of order $p$. In particular, we show how to use the existing lowerbounds for the estimation of the cumulative distribution function in dimensionone to find lower bounds for the Wasserstein deconvolution in any dimension.

Minimax rates of convergence for Wasserstein deconvolution with supersmooth errors in any dimension