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We point out some pitfalls related to the concept of an oracle property as used in Fan and Li (2001, 2002, 2004) which are reminiscent of the well-known pitfalls related to Hodges’ estimator. The oracle property is often a consequence of sparsity of an estimator. We show that any estimator satisfying a sparsity property has maximal risk that converges to the supremum of the loss function; in particular, the maximal risk diverges to inﬁnity when ever the loss function is unbounded. For ease of presentation the result is set in the framework of a linear regression model, but generalizes far beyond that setting. In a Monte Carlo study we also assess the extent of the problem inﬁnite samples for the smoothly clipped absolute deviation (SCAD) estimator introduced in Fan and Li (2001). We ﬁnd that this estimator can perform rather poorly inﬁnite samples and that its worst-case performance relative to maximum likelihood deteriorates with increasing sample size when the estimator is tuned to sparsity.
Leeb, Hannes and Pötscher, Benedikt M., "Sparse Estimators and the Oracle Property, or the Return of Hodges’ Estimator" (2005). Cowles Foundation Discussion Papers. 1782.