On Hodges' Superefficiency and Merits of Oracle Property in Model Selection

Abstract: The oracle property of model selection procedures has attracted a large volume of favorable publications in the literature, but also faced criticisms of being ineffective and misleading in applications. In this paper, we introduce a class of estimators that can easily produce model selection procedures possessing the oracle property and discuss the merits of the oracle property by analyzing the performance of such estimators in finite sample size theoretically. Specifically, we propose a new type of Hodges' estimators capable of reducing the asymptotic variance of any given estimator over a multi-dimensional subspace of the parameter space, which can easily produce model selection procedures with the oracle and some other desired properties. This new type of oracle estimators, however, perform poorly at some values of the parameters for estimation, and there is no convincing reason to declare that oracle estimators are better than the traditional estimators such as the MLE and LSE. Consequently, the merits of the oracle property for model selection as claimed in the literature are probably grossly overstated, and the criticisms of the oracle property are justifiable.