Autore
Scricciolo, Catia

Titolo
Bayes and maximum likelihood for L1-Wasserstein deconvolution of Laplace mixtures
Periodico
Statistical methods & applications : Journal of the Italian Statistical Society
Anno: 2018 - Volume: 27 - Fascicolo: 2 - Pagina iniziale: 333 - Pagina finale: 362

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 L2-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−3/8log1/8n rate for the maximum likelihood estimator. Then, appealing to an inversion inequality translating the L2-norm and the Hellinger distance between general kernel mixtures, with a kernel density having polynomially decaying Fourier transform, into any Lp-Wasserstein distance, 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 L1-Wasserstein metric for the Bayes’ and maximum likelihood estimators of the mixing distribution. Merging in the L1-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.



SICI: 1618-2510(2018)27:2<333:BAMLFL>2.0.ZU;2-5

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