Sparse estimators and the oracle property, or the return of Hodges' estimator

We point out some pitfalls related to the concept of an oracle property as used in Fan and Li [2001. Variable selection via nonconcave penalized likelihood and its oracle properties. Journal of the American Statistical Association 96, 1348-1360; 2002. Variable selection for Cox's proportional hazards model and frailty model. Annals of Statistics 30, 74-99; 2004. New estimation and model selection procedures for semiparametric modeling in longitudinal data analysis. Journal of the American Statistical Association 99, 710-723] 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 whenever 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 in finite samples for the smoothly clipped absolute deviation (SCAD) estimator introduced in Fan and Li [2001. Variable selection via nonconcave penalized likelihood and its oracle properties. Journal of the American Statistical Association 96, 1348-1360]. We find that this estimator can perform rather poorly in finite samples and that its worst-case performance relative to maximum likelihood deteriorates with increasing sample size when the estimator is tuned to sparsity.