Bayes and maximum likelihood for $$L^1$$ L 1 -Wasserstein deconvolution of Laplace mixtures

We consider the problem of recovering a distribution function on the real line from observations additively contaminated with errors following the standard Laplace distribution. Assuming that the latent distribution is completely unknown leads to a nonparametric deconvolution problem. We begin by studying the rates of convergence relative to the $$L^2$$ L 2 -norm and the Hellinger metric for the direct problem of estimating the sampling density, which is a mixture of Laplace densities with a possibly unbounded set of locations: the rate of convergence for the Bayes’ density estimator corresponding to a Dirichlet process prior over the space of all mixing distributions on the real line matches, up to a logarithmic factor, with the $$n^{-3/8}\log ^{1/8}n$$ n - 3 / 8 log 1 / 8 n rate for the maximum likelihood estimator. Then, appealing to an inversion inequality translating the $$L^2$$ L 2 -norm and the Hellinger distance between general kernel mixtures, with a kernel density having polynomially decaying Fourier transform, into any $$L^p$$ L p -Wasserstein distance, $$p\ge 1$$ p ≥ 1 , between the corresponding mixing distributions, provided their Laplace transforms are finite in some neighborhood of zero, we derive the rates of convergence in the $$L^1$$ L 1 -Wasserstein metric for the Bayes’ and maximum likelihood estimators of the mixing distribution. Merging in the $$L^1$$ L 1 -Wasserstein distance between Bayes and maximum likelihood follows as a by-product, along with an assessment on the stochastic order of the discrepancy between the two estimation procedures.