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

A. Chatterjee, S. N. Lahiri
2013 Annals of Statistics  
Zou [J. Amer. Statist. Assoc. 101 (2006) 1418-1429] proposed the Adaptive LASSO (ALASSO) method for simultaneous variable selection and estimation of the regression parameters, and established its oracle property. In this paper, we investigate the rate of convergence of the ALASSO estimator to the oracle distribution when the dimension of the regression parameters may grow to infinity with the sample size. It is shown that the rate critically depends on the choices of the penalty parameter and
more » ... he initial estimator, among other factors, and that confidence intervals (CIs) based on the oracle limit law often have poor coverage accuracy. As an alternative, we consider the residual bootstrap method for the ALASSO estimators that has been recently shown to be consistent; cf. Chatterjee and Lahiri [J. Amer. Statist. Assoc. 106 (2011a) 608-625]. We show that the bootstrap applied to a suitable studentized version of the ALASSO estimator achieves second-order correctness, even when the dimension of the regression parameters is unbounded. Results from a moderately large simulation study show marked improvement in coverage accuracy for the bootstrap CIs over the oracle based CIs.
doi:10.1214/13-aos1106 fatcat:edoizpwjdrbt5jfho4hf62dwvi