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.Comment: 20 pages, 4 figure