 Solution

One standard construction uses the Debiased Lasso. Starting from the Square-root Lasso estimate $\hat{\beta }$, estimate a vector ${\hat{\Theta }}_j$ that approximately inverts the $j$th column of the empirical Gram matrix $\hat{\Sigma }=X^{T}X/n$, for example by a Nodewise Lasso. Defineand estimate $\sigma$ by $\hat{\sigma }=\Vert Y-X\hat{\beta }{\Vert }_2/\sqrt{n}$. The approximate two-sided level-$\alpha$ test rejects $H_j^{\beta }$ whenwhere $z_{1-\alpha /2}$ is a standard normal quantile.

Sufficient high-dimensional conditions include a Compatibility condition for the Lasso bounded away from zero, $\gamma \asymp \sqrt{\log p/n}$, $\log p=o(n)$, andtogether with the corresponding sparsity and consistency conditions for the nodewise inverse-Gram estimate. Under these assumptions the Debiased-Lasso asymptotic normality makes the rejection probability under $H_j^{\beta }$ tend to $\alpha$.