  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 the 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.
