Rates of convergence of the Adaptive LASSO estimators to the Oracle distribution and higher order refinements by the bootstrap

摘要

Zou [J. Amer. Statist. Assoc. 101 (2006) 1418-1429] proposed the AdaptiveLASSO (ALASSO) method for simultaneous variable selection and estimation of theregression parameters, and established its oracle property. In this paper, weinvestigate the rate of convergence of the ALASSO estimator to the oracledistribution when the dimension of the regression parameters may grow toinfinity with the sample size. It is shown that the rate critically depends onthe choices of the penalty parameter and the initial estimator, among otherfactors, and that confidence intervals (CIs) based on the oracle limit lawoften have poor coverage accuracy. As an alternative, we consider the residualbootstrap method for the ALASSO estimators that has been recently shown to beconsistent; cf. Chatterjee and Lahiri [J. Amer. Statist. Assoc. 106 (2011a)608-625]. We show that the bootstrap applied to a suitable studentized versionof the ALASSO estimator achieves second-order correctness, even when thedimension of the regression parameters is unbounded. Results from a moderatelylarge simulation study show marked improvement in coverage accuracy for thebootstrap CIs over the oracle based CIs.