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 known ordinary smooth error distributions. For locally Hölder continuous or Sobolev regular mixing densities, a bona fide distribution function estimator recently proposed in the literature is shown to achieve minimax-optimal convergence rates (up to logarithmic factors) for all values of the index $$\beta >0$$ β > 0 of the Fourier transform of the error distribution under the 1-Wasserstein distance. However, for p-Wasserstein metrics of any order $$p>1$$ p > 1 , these rates are known to be minimax-optimal only for $$\varvec{\beta }\le {1}/{2}$$ β ≤ 1 / 2 . Using the representation of the p-Wasserstein distance between two probability measures as the $$L^p$$ L p -distance between their corresponding quantile functions, we propose an estimator defined as the approximate minimizer of the $$L^p$$ L p -distance from a minimum-contrast quantile function estimator. This estimator is based on the integrated classical deconvolution kernel density estimator and achieves minimax-optimal rates (up to logarithmic factors) for any $$\beta >0$$ β > 0 and Hölder continuous densities on some bounded interval of quantiles. The result fills an important gap in the literature by, firstly, establishing previously unknown minimax-optimal convergence rates and, secondly, showing that these rates depend only on intrinsic elements of the decision problem. Specifically, they are determined by the class parameters describing the regularity of the mixing and error densities and are independent of the choice of the loss function.