Small Tuning Parameter Selection for the Debiased Lasso

Abstract

In this study, we investigate the bias and variance properties of the debiased Lasso in linear regression when the tuning parameter of the node-wise Lasso is selected to be smaller than in previous studies. We consider the case where the number of covariates p is bounded by a constant multiple of the sample size n. First, we show that the bias of the debiased Lasso can be reduced without diverging the asymptotic variance by setting the order of the tuning parameter to (Formula presented.). This implies that the debiased Lasso has asymptotic normality provided that the number of nonzero coefficients (Formula presented.) satisfies (Formula presented.), whereas previous studies require (Formula presented.) if no sparsity assumption is imposed on the inverse of the second moment matrix of covariates. Second, we propose a data-driven tuning parameter selection procedure for the node-wise Lasso that is consistent with our theoretical results. Simulation studies show that our procedure yields confidence intervals with good coverage properties in various settings. We also present a real economic data example to demonstrate the efficacy of our selection procedure.