Distributional Consistency of Lasso by Perturbation Bootstrap

TL;DR

This paper introduces a perturbation bootstrap method for the Lasso estimator in heteroscedastic linear regression, valid for both random and non-random covariates, with demonstrated finite-sample effectiveness.

Contribution

It develops and validates a novel perturbation bootstrap approach for Lasso distribution approximation applicable regardless of covariate nature.

Findings

Bootstrap method accurately approximates Lasso distribution in simulations.

Method is valid for heteroscedastic errors and both covariate types.

Simulation results support finite-sample effectiveness.

Abstract

Least Absolute Shrinkage and Selection Operator or the Lasso, introduced by Tibshirani (1996), is a popular estimation procedure in multiple linear regression when underlying design has a sparse structure, because of its property that it sets some regression coefficients exactly equal to 0. In this article, we develop a perturbation bootstrap method and establish its validity in approximating the distribution of the Lasso in heteroscedastic linear regression. We allow the underlying covariates to be either random or non-random. We show that the proposed bootstrap method works irrespective of the nature of the covariates, unlike the resample-based bootstrap of Freedman (1981) which must be tailored based on the nature (random vs non-random) of the covariates. Simulation study also justifies our method in finite samples.