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April 2020 On the nonparametric maximum likelihood estimator for Gaussian location mixture densities with application to Gaussian denoising
Sujayam Saha, Adityanand Guntuboyina
Ann. Statist. 48(2): 738-762 (April 2020). DOI: 10.1214/19-AOS1817

Abstract

We study the nonparametric maximum likelihood estimator (NPMLE) for estimating Gaussian location mixture densities in $d$-dimensions from independent observations. Unlike usual likelihood-based methods for fitting mixtures, NPMLEs are based on convex optimization. We prove finite sample results on the Hellinger accuracy of every NPMLE. Our results imply, in particular, that every NPMLE achieves near parametric risk (up to logarithmic multiplicative factors) when the true density is a discrete Gaussian mixture without any prior information on the number of mixture components. NPMLEs can naturally be used to yield empirical Bayes estimates of the oracle Bayes estimator in the Gaussian denoising problem. We prove bounds for the accuracy of the empirical Bayes estimate as an approximation to the oracle Bayes estimator. Here our results imply that the empirical Bayes estimator performs at nearly the optimal level (up to logarithmic factors) for denoising in clustering situations without any prior knowledge of the number of clusters.

Citation

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Sujayam Saha. Adityanand Guntuboyina. "On the nonparametric maximum likelihood estimator for Gaussian location mixture densities with application to Gaussian denoising." Ann. Statist. 48 (2) 738 - 762, April 2020. https://doi.org/10.1214/19-AOS1817

Information

Received: 1 December 2017; Revised: 1 October 2018; Published: April 2020
First available in Project Euclid: 26 May 2020

zbMATH: 07241567
MathSciNet: MR4102674
Digital Object Identifier: 10.1214/19-AOS1817

Subjects:
Primary: 62C10, 62C12, 62G07

Rights: Copyright © 2020 Institute of Mathematical Statistics

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Vol.48 • No. 2 • April 2020
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