On the asymptotic variance of the debiased Lasso

Abstract

We consider the high-dimensional linear regression model $Y=X\beta^{0}+\epsilon$ with Gaussian noise $\epsilon$ and Gaussian random design $X$. We assume that $\Sigma:=\mathrm{I\hskip-0.48emE}X^{T}X/n$ is non-singular and write its inverse as $\Theta :=\Sigma^{-1}$. The parameter of interest is the first component $\beta_{1}^{0}$ of $\beta^{0}$. We show that in the high-dimensional case the asymptotic variance of a debiased Lasso estimator can be smaller than $\Theta_{1,1}$. For some special such cases we establish asymptotic efficiency. The conditions include $\beta^{0}$ being sparse and the first column $\Theta_{1}$ of $\Theta$ being not sparse. These sparsity conditions depend on whether $\Sigma$ is known or not.