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We consider the problem of recovering a distribution function on the realline from observations additively contaminated with errors following thestandard Laplace distribution. Assuming that the latent distribution iscompletely unknown leads to a nonparametric deconvolution problem. We begin bystudying the rates of convergence relative to the $L^2$-norm and the Hellingermetric for the direct problem of estimating the sampling density, which is amixture of Laplace densities with a possibly unbounded set of locations: therate of convergence for the Bayes' density estimator corresponding to aDirichlet process prior over the space of all mixing distributions on the realline matches, up to a logarithmic factor, with the $n^{-3/8}\log^{1/8}n$ ratefor the maximum likelihood estimator. Then, appealing to an inversioninequality translating the $L^2$-norm and the Hellinger distance betweengeneral kernel mixtures, with a kernel density having polynomially decayingFourier transform, into any $L^p$-Wasserstein distance, $p\geq1$, between thecorresponding mixing distributions, provided their Laplace transforms arefinite in some neighborhood of zero, we derive the rates of convergence in the$L^1$-Wasserstein metric for the Bayes' and maximum likelihood estimators ofthe mixing distribution. Merging in the $L^1$-Wasserstein distance betweenBayes and maximum likelihood follows as a by-product, along with an assessmenton the stochastic order of the discrepancy between the two estimationprocedures.

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