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Selection of important covariates and to drop the unimportant ones from ahigh-dimensional regression model is a long standing problem and hence havereceived lots of attention in the last two decades. After selecting the correctmodel, it is also important to properly estimate the existing parameterscorresponding to important covariates. In this spirit, Fan and Li (2001)proposed Oracle property as a desired feature of a variable selection method.Oracle property has two parts; one is the variable selection consistency (VSC)and the other one is the asymptotic normality. Keeping VSC fixed and making theother part stronger, Fan and Lv (2008) introduced the strong oracle property.In this paper, we consider different penalized regression techniques which areVSC and classify those based on oracle and strong oracle property. We show thatboth the residual and the perturbation bootstrap methods are second ordercorrect for any penalized estimator irrespective of its class. Most interestingof all is the Lasso, introduced by Tibshirani (1996). Although Lasso is VSC, itis not asymptotically normal and hence fails to satisfy the oracle property.

Higher Order Refinements by Bootstrap in Lasso and other Penalized Regression Methods