Optimal Shrinkage of Eigenvalues in the Spiked Covariance Model

• Activity Number: 336
• Type: Invited
• Date/Time: Tuesday, August 11, 2015 : 10:30 AM to 12:20 PM
• Sponsor: IMS
• Author(s): Iain Johnstone* and David Donoho and Matan Gavish
• Companies: Stanford University and Stanford University and Stanford University
• Keywords: covariance estimation ; optimal shrinkage ; stein loss ; entropy loss ; spiked covariance

Abstract

Stein found that shrinking the eigenvalues of the empirical covariance matrix makes a better estimator for the population covariance. We consider a framework for high-dimensional covariance estimation, based on the spiked covariance model, in which a single scalar nonlinearity is applied individually to each of the eigenvalues of the empirical covariance matrix. We show, for a variety of loss functions, that there is a unique admissible shrinker, often explicit, that dominates all other shrinkers asymptotically, and consider some related optimality properties.