Shrinkage priors for single-spiked covariance models

Abstract

Shrinkage estimation of eigenvalues of covariance matrices for Gaussian spiked covariance models are known to be effective especially in high-dimensional problems. Various shrinkage methods have been proposed, and these typically employ nonlinear functions of eigenvalues of sample covariance matrices. Here we investigate Bayesian shrinkage methods for single-spiked covariance models. The choice of priors and the construction of the Bayes estimator are considered, and it is proved that the Bayes estimator of the present shrinkage prior dominates asymptotically that of the Jeffreys prior regarding the Kullback–Leibler risk. The Bayes estimators are obtained as the posterior mean of the covariance matrices. In numerical simulations, the Bayes methods based on the present shrinkage prior and the Jeffreys prior are compared with other non-Bayes methods.