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

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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Statistical Methods & Applications Springer Journals

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

, Volume 27 (2) – Sep 15, 2017
30 pages

/lp/springer_journal/bayes-and-maximum-likelihood-for-l-1-l-1-wasserstein-deconvolution-of-RfaLwspclq
Publisher
Springer Berlin Heidelberg
Subject
Statistics; Statistics, general; Statistical Theory and Methods; Statistics for Business/Economics/Mathematical Finance/Insurance; Statistics for Engineering, Physics, Computer Science, Chemistry and Earth Sciences; Statistics for Life Sciences, Medicine, Health Sciences; Statistics for Social Science, Behavorial Science, Education, Public Policy, and Law
ISSN
1618-2510
eISSN
1613-981X
D.O.I.
10.1007/s10260-017-0400-4
Publisher site
See Article on Publisher Site

### Abstract

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.

### Journal

Statistical Methods & ApplicationsSpringer Journals

Published: Sep 15, 2017

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