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 Rd 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.