A nonparametric empirical Bayes approach to covariance matrix estimation

We propose an empirical Bayes method to estimate high-dimensional covariance matrices. Our procedure centers on vectorizing the covariance matrix and treating matrix estimation as a vector estimation problem. Drawing from the compound decision theory literature, we introduce a new class of decision rules that generalizes several existing procedures. We then use a nonparametric empirical Bayes g-modeling approach to estimate the oracle optimal rule in that class. This allows us to let the data itself determine how best to shrink the estimator, rather than shrinking in a pre-determined direction such as toward a diagonal matrix. Simulation results and a gene expression network analysis shows that our approach can outperform a number of state-of-the-art proposals in a wide range of settings, sometimes substantially.