PERFORMANCE LIMITS FOR ESTIMATORS OF THE RISK OR DISTRIBUTION OF SHRINKAGE-TYPE ESTIMATORS,AND SOME GENERAL LOWER RISK-BOUND RESULTS

摘要

We consider the problem of estimating measures of precision of shrinkage-type estimators such as their risk or distribution.The notion of shrinkage-type estimators here refers to estimators such as the James-Stein estimator and Lasso-type estimators,in addition to "thresholding" estimators such as,e.g.,Hodges' so-called superefficient estimator.Although 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 obtain in the paper.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 also obtained in the paper.