Shrinkage estimation of large covariance matrices: Keep it simple, statistician?

Under rotation-equivariant decision theory, sample covariance matrix eigenvalues can be optimally shrunk by recombining eigenvectors with a (potentially nonlinear) function of the unobservable population matrix. The optimal shape this reflects loss/risk that is to minimized. We solve problem estimation under variety loss functions motivated statistical precedent, probability and differential geometry. A key ingredient our nonlinear shrinkage methodology new estimator angle between eigenvectors, without making strong assumptions on eigenvalues. also introduce broad family estimators handle all regular functional transformations large-dimensional asymptotics. In addition, we compare via Monte Carlo simulations two simpler ones from literature, linear based spiked model.