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August 2019 Perturbation bootstrap in adaptive Lasso
Debraj Das, Karl Gregory, S. N. Lahiri
Ann. Statist. 47(4): 2080-2116 (August 2019). DOI: 10.1214/18-AOS1741

Abstract

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 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.

Citation

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Debraj Das. Karl Gregory. S. N. Lahiri. "Perturbation bootstrap in adaptive Lasso." Ann. Statist. 47 (4) 2080 - 2116, August 2019. https://doi.org/10.1214/18-AOS1741

Information

Received: 1 November 2016; Revised: 1 February 2018; Published: August 2019
First available in Project Euclid: 21 May 2019

zbMATH: 07082280
MathSciNet: MR3953445
Digital Object Identifier: 10.1214/18-AOS1741

Subjects:
Primary: 62J07
Secondary: 62E20 , 62G09

Keywords: Alasso , modified perturbation bootstrap , naive perturbation bootstrap , oracle , second-order correctness

Rights: Copyright © 2019 Institute of Mathematical Statistics

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Vol.47 • No. 4 • August 2019
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