We consider Bayesian nonparametric density estimation with a Dirichlet process kernel mixture as a prior on the class of Lebesgue univariate densities, the emphasis being on the achievability of the error rate n^{-1/2}, up to a logarithmic factor, depending upon the kernel. We derive rates of convergence for the Bayes’ estimator of super-smooth densities that are location-scale mixtures of densities whose Fourier transforms have sub-exponential tails. We show that a nearly parametric rate is attainable in the L^1-norm, under weak assumptions on the tail decay of the true mixing distribution and the overall Dirichlet process base measure.
Rates for Bayesian Estimation of Location-Scale Mixtures of Super-Smooth Densities
Scricciolo, Catia
2016-01-01
Abstract
We consider Bayesian nonparametric density estimation with a Dirichlet process kernel mixture as a prior on the class of Lebesgue univariate densities, the emphasis being on the achievability of the error rate n^{-1/2}, up to a logarithmic factor, depending upon the kernel. We derive rates of convergence for the Bayes’ estimator of super-smooth densities that are location-scale mixtures of densities whose Fourier transforms have sub-exponential tails. We show that a nearly parametric rate is attainable in the L^1-norm, under weak assumptions on the tail decay of the true mixing distribution and the overall Dirichlet process base measure.
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