Optimal Shrinkage of Eigenvalues in the Spiked Covariance Model

Iain Johnstone, Stanford U.

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. Joint work with David Donoho and Matan Gavish.