Minimax rates for Wasserstein deconvolution of regular distributions with ordinary smooth errors

We study the problem of univariate distribution function estimation with respect to Wasserstein metrics in nonparametric deconvolution models with ordinary smooth error distributions. When the error density is known, for H\"older or Sobolev regular mixing distributions, a recently proposed isotone distribution function estimator is shown to attain minimax-optimal (up to logarithmic factors) convergence rates for all values of the index $\beta>0$ of the Fourier transform of the error distribution under the $1$-Wasserstein metric, while, for $p$-Wasserstein metrics of any order $p>1$, these rates are known to be minimax-optimal only for $\boldsymbol\beta\leq{1}/{2}$. Using the representation of the $p$-Wasserstein metric between two probability measures as the $L^p$-distance between the corresponding quantile functions, an estimator defined as the approximate minimizer of the $L^p$-distance from a minimum contrast quantile function estimator based on the integrated classical deconvolution kernel density estimator, yields minimax-optimal (up to log-factors) rates for any $\beta>0$ over locally H\"older regular densities in some bounded interval of quantiles. The result fills an important gap in the literature, firstly establishing previously unknown minimax-optimal convergence rates and, secondly, showing that these rates only depend on intrinsic elements of the decision problem, namely, the parameters describing the regularity of the classes of the mixing and error densities, and not on the loss function.