Performance Limits for Estimators of the Risk or Distribution of Shrinkage-Type Estimators, and Some General Lower Risk-Bound Results

Abstract

We consider the problem of estimating measures of precision of shrinkage-type estimators like their risk or distribution. The notion of shrinkage-type estimators here refers to estimators like the James-Stein estimator or Lasso-type estimators, as well as to 'thresholding' estimators such as, e.g., Hodges' so-called superefficient estimator. While the precision measures of such estimators typically can be estimated consistently, we show that they cannot be estimated uniformly consistently (even locally). This follows as a corollary to (locally) uniform lower bounds on the performance of estimators of the precision measures that we also obtain. These lower bounds are typically quite large (e.g., they approach 1/2 or 1 depending on the situation considered). The analysis is based on some general lower risk bounds and related general results on the (non)existence of uniformly consistent estimators.