  The Adaptive Lasso(Alasso) was proposed by Zou [\textit{J. Amer. Statist.
Assoc. \textbf{101} (2006) 1418-1429}] as a modification of the Lasso for the
purpose of simultaneous variable selection and estimation of the parameters in
a linear regression model. Zou (2006) established that the Alasso estimator is
variable-selection consistent as well as asymptotically Normal in the indices
corresponding to the nonzero regression coefficients in certain
fixed-dimensional settings. In an influential paper, Minnier, Tian and Cai
[\textit{J. Amer. Statist. Assoc. \textbf{106} (2011) 1371-1382}] proposed a
perturbation bootstrap method and established its distributional consistency
for the Alasso estimator in the fixed-dimensional setting. In this paper,
however, we show that this (naive) perturbation bootstrap fails to achieve
second order correctness in approximating the distribution of the Alasso
estimator. We propose a modification to the perturbation bootstrap objective
function and show that a suitably studentized version of our modified
perturbation bootstrap Alasso estimator achieves second-order correctness even
when the dimension of the model is allowed to grow to infinity with the sample
size. As a consequence, inferences based on the modified perturbation bootstrap
will be more accurate than the inferences based on the oracle Normal
approximation. We give simulation studies demonstrating good finite-sample
properties of our modified perturbation bootstrap method as well as an
illustration of our method on a real data set.
