On the asymptotic variance of the debiased Lasso

Abstract

We consider the high-dimensional linear regression model Y = XΒ0 +ε with Gaussian noise ε and Gaussian random design X. We assume that Σ:= IEXT X/n is non-singular and write its inverse as θ:= Σ-1. The parameter of interest is the first component Β01 of Β0. We show that in the high-dimensional case the asymptotic variance of a debiased Lasso estimator can be smaller than θ1,1. For some special such cases we establish asymptotic efficiency. The conditions include Β0 being sparse and the first column θ1 of θ being not sparse. These sparsity conditions depend on whether Σ is known or not.