Yale University
Department of Statistics
Seminar

Monday, October 21, 2002

        On Inference with Shrinkage-Type Estimators
 (and some general results on consistent vs. uniformly consistent estimators)

AUTHORS:
        Hannes Leeb and Benedikt M. Poetscher

We are concerned with the estimation of quantities like the distribution or the risk of shrinkage-type estimators including, e.g., Lasso-type estimators, Bridge estimators, the James-Stein estimator, or Hodges' superefficient estimator.  While these quantities can be estimated consistently, we show that they cannot be estimated in a uniformly consistent fashion (even in a local sense). We provide finite-sample lower bounds for the estimation error probability.  In large samples, these bounds typically approach 1/2 or even one.  To analyze these phenomena, we also provide some general results on the (non-) existence of uniformly consistent estimators.