COWLES FOUNDATION FOR RESEARCH IN ECONOMICS
AT YALE UNIVERSITY

Box 208281
New Haven, CT 06520-8281

COWLES FOUNDATION DISCUSSION PAPER NO. 1500

Sparse Estimators and the Oracle Property, or the Return of Hodges' Estimator

Hannes Leeb
Department of Statistics, Yale University
and
Benedikt M. Pötscher
Department of Statistics, University of Vienna

February 2005
Revised April 2007

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.

AMS 2000 Subject Classifications: Primary 62J07, 62C99; secondary 62E20, 62F10, 62F12

JEL Classifications: C20, C51

Keywords and phrases: Oracle property, Sparsity, Penalized maximum likelihood, Penalized least squares, Hodges’ estimator, SCAD, Lasso, Bridge estimator, Hard-thresholding, Maximal risk, Maximal absolute bias, Non-uniform limits