ON THE NONPARAMETRIC MAXIMUM LIKELIHOOD ESTIMATOR FOR GAUSSIAN LOCATION MIXTURE DENSITIES WITH APPLICATION TO GAUSSIAN DENOISING

• 成果类型: Article
• 署名作者: Saha, Sujayam; Guntuboyina, Adityanand
• 署名单位: University of California System; University of California Berkeley
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/19-AOS1817
• 发表日期: 2020
• 页码: 738-762
• 关键词: Empirical Bayes; CONVERGENCE; algorithm; selection; geometry; vector; rates

摘要

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.