Linear shrinkage estimation of high-dimensional means

Abstract

In estimation of a mean vector, consider the case that the mean vector is suspected to be in one or two general linear subspaces. Then it is reasonable to shrink a sample mean vector toward the restricted estimators on the linear subspaces. Motivated from a standard Bayesian argument, we propose single and double shrinkage estimators in which their optimal weights are estimated consistently in high dimension without assuming any specific distributions. Asymptotic relative improvement in risk of shrinkage estimators over the sample mean vector is derived in high dimension, and it is shown that the gain in improvement by shrinkage depends on the linear subspace. Finally, the performance of the linear shrinkage estimators is numerically investigated by simulation.