  Covariance matrix estimation is a fundamental statistical task in many
applications, but the sample covariance matrix is sub-optimal when the sample
size is comparable to or less than the number of features. Such
high-dimensional settings are common in modern genomics, where covariance
matrix estimation is frequently employed as a method for inferring gene
networks. To achieve estimation accuracy in these settings, existing methods
typically either assume that the population covariance matrix has some
particular structure, for example sparsity, or apply shrinkage to better
estimate the population eigenvalues. In this paper, we study a new approach to
estimating high-dimensional covariance matrices. We first frame covariance
matrix estimation as a compound decision problem. This motivates defining a
class of decision rules and using a nonparametric empirical Bayes g-modeling
approach to estimate the optimal rule in the class. Simulation results and gene
network inference in an RNA-seq experiment in mouse show that our approach is
comparable to or can outperform a number of state-of-the-art proposals.
