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We study the Nonparametric Maximum Likelihood Estimator (NPMLE) forestimating Gaussian location mixture densities in $d$-dimensions fromindependent observations. Unlike usual likelihood-based methods for fittingmixtures, NPMLEs are based on convex optimization. We prove finite sampleresults on the Hellinger accuracy of every NPMLE. Our results imply, inparticular, that every NPMLE achieves near parametric risk (up to logarithmicmultiplicative factors) when the true density is a discrete Gaussian mixturewithout any prior information on the number of mixture components. NPMLEs cannaturally be used to yield empirical Bayes estimates of the Oracle Bayesestimator in the Gaussian denoising problem. We prove bounds for the accuracyof the empirical Bayes estimate as an approximation to the Oracle Bayesestimator. Here our results imply that the empirical Bayes estimator performsat nearly the optimal level (up to logarithmic multiplicative factors) fordenoising in clustering situations without any prior knowledge of the number ofclusters.

On the nonparametric maximum likelihood estimator for Gaussian location mixture densities with application to Gaussian denoising