Perturbation bootstrap in adaptive lasso

The Adaptive Lasso (Alasso) was proposed by Zou [J. Amer. Statist. Assoc. 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 [J. Amer. Statist. Assoc. 101 (2006) 1418-1429] 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 [J. Amer. Statist. Assoc. 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 secondorder 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.