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Problems involving the simultaneous estimation of multiple parameters arise in many areas of theoretical and applied statistics. A canonical example is the estimation of a vector of normal means. Frequently, structural information about relationships between the parameters of interest is available. For example, in a gene expression denoising problem, genes with similar functions may have similar expression levels. Despite its importance, structural information has not been well-studied in the simultaneous estimation literature, perhaps in part because it poses challenges to the usual geometric or empirical Bayes shrinkage estimation paradigms. This article proposes that some of these challenges can be resolved by adopting an alternate paradigm, based on regression modeling. This approach can naturally incorporate structural information and also motivates new shrinkage estimation and inference procedures. As an illustration, this regression paradigm is used to develop a class of estimators with asymptotic risk optimality properties that perform well in simulations and in denoising gene expression data from a single cell RNA-sequencing experiment. [ABSTRACT FROM AUTHOR]

We propose new nonparametric empirical Bayes methods for high-dimensional classification. Our classifiers are designed to approximate the Bayes classifier in a hypothesized hierarchical model, where the prior distributions for the model parameters are estimated nonparametrically from the training data. As is common with nonparametric empirical Bayes, the proposed classifiers are effective in high-dimensional settings even when the underlying model parameters are in fact nonrandom. We use nonparametric maximum likelihood estimates of the prior distributions, following the elegant approach studied by Kiefer & Wolfowitz in the 1950s. However, our implementation is based on a recent convex optimization framework for approximating these estimates that is well-suited for large-scale problems. We derive new theoretical results on the accuracy of the approximate estimator, which help control the misclassification rate of one of our classifiers. We show that our methods outperform several existing methods in simulations and perform well when gene expression microarray data is used to classify cancer patients. [ABSTRACT FROM AUTHOR]