Bootstrapping Lasso Estimators

In this article, we consider bootstrapping the Lasso estimator of the regression parameter in a multiple linear regression model. It is known that the standard bootstrap method fails to be consistent. Here, we propose a modified bootstrap method, and show that it provides valid approximation to the distribution of the Lasso estimator, for all possible values of the unknown regression parameter vector, including the case where some of the components are zero. Further, we establish consistency of the modified bootstrap method for estimating the asymptotic bias and variance of the Lasso estimator. We also show that the residual bootstrap can be used to consistently estimate the distribution and variance of the adaptive Lasso estimator. Using the former result, we formulate a novel data-based method for choosing the optimal penalizing parameter for the Lasso using the modified bootstrap. A numerical study is performed to investigate the finite sample performance of the modified bootstrap. The methodology proposed in the article is illustrated with a real data example.