Performance limits for estimators of the risk or distribution of shrinkage-type estimators, and some general lower risk-bound results

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.