期刊
ANNALS OF STATISTICS
卷 48, 期 2, 页码 738-762出版社
INST MATHEMATICAL STATISTICS-IMS
DOI: 10.1214/19-AOS1817
关键词
Density estimation; Gaussian mixture model; convex optimization; Hellinger distance; metric entropy; rate of convergence; model selection; adaptive estimation; convex clustering
资金
- NSF CAREER [DMS-16-54589]
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.
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