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

The subject of this paper is the estimation of a probability measure on R d from the data observed with an additive noise, under the Wasserstein metric of order p (with p ≥ 1 ). We assume that the distribution of the errors is known and belongs to a class of supersmooth distributions, and we give optimal rates of convergence for the Wasserstein metric of order p . In particular, we show how to use the existing lower bounds for the estimation of the cumulative distribution function in dimension one to find lower bounds for the Wasserstein deconvolution in any dimension.