Stein Estimation in High Dimensions and the Bootstrap'

The Stein estimator 's and the better positive-part Stein estimator gpS both dominate the sample mean, under quadratic loss, in the N(g, I) model of dimension q > 3. Standard large sample theory does not explaill this phenomenon well. Plausible bootstrap estimators for the risk of 's do not converge correctly at the shrinkage point as sample size n increases. By analyzing a submodel exactly, with the help of results from directional statistics, and then letting dimension q -* oo, we find: * In high dimensions, (s and &S are approximately admissible and approximately minimax on large compact balls about the shrinkage point. The sample mean is neither. * A new estimator of (, asymptotically equivalent to &pS as q -* oo, appears to dominate &Ps slightly. * Resampling from a N(t, I) distribution, where k112 estimates ItI2 well, is the key to consistent bootstrap risk estimation for orthogonally equivariant estimators of (. Choosing ( to be the Stein estimator or the positive-part Stein!estimator or the sample mean does not work. * Estimators of 141 are subject to a sharp local asymptotic minimax bound as q increases.