Debiased Lasso and confidence intervals

Let $G=X^\top X/n$ and choose a matrix $M$ depending on the design but not on the observation noise after conditioning on that design. Define the corrected vector by adding $MX^\top(y-X\widehat\beta)/n$ to an initial Lasso estimate. Substitute the linear model and collect terms. The corrected error splits exactly into a linear transform of the noise and the original coefficient error multiplied by $I-MG$. No approximation enters this algebraic identity, and no assumption that the selected support is correct is needed.

When the design is wide, its Gram matrix is singular, so a global inverse cannot exist. The correction instead seeks small entrywise inverse error or suitably accurate rows. Nodewise regression is one construction, but its success requires additional design and sparsity assumptions; it is not justified by sparsity of the original coefficient alone. Conditional independence from the observation noise matters for the Gaussian calculation. If a proposed correction is tuned using the same noise in an uncontrolled way, its leading term need not have the claimed conditional distribution.

Use the maximum absolute entry norm for $I-MG$, rather than its induced row-sum norm. Hölder’s inequality bounds each remainder coordinate by that entrywise error times the absolute coefficient-error norm. With independent Gaussian noise of covariance $\sigma^2I_n$, the leading coordinate has conditional variance $\sigma^2(MGM^\top)_{jj}$. If this variance is bounded away from zero, the standardized remainder tends to zero in probability, and the noise-scale estimate is ratio-consistent, a fixed predetermined coordinate has an asymptotically standard normal pivot. Take a fixed confidence error level strictly between zero and one for the interval below.

For illustration, inverse error of order $\sqrt{\log p/n}$ and absolute coefficient error of order $s\sqrt{\log p/n}$ yield a scaled remainder of order $s\log p/\sqrt n$, provided the design constants and noise scale remain controlled. Its vanishing is a stronger requirement than the prediction scale $s\log p/n\to0$. These are conditional rate calculations, not a theorem guaranteeing those inverse and coefficient bounds for every design. Nodewise precision sparsity or another valid construction must supply the missing conditions.

A fixed-coordinate coverage statement concerns a coordinate chosen independently of the observed responses. Selecting the most impressive corrected coefficient afterward is a different target, and applying many marginal intervals does not create joint coverage. Multiplicity correction or a justified approximation to the maximum is needed for simultaneous claims. Uniformly honest coverage over a parameter class also requires the remainder and variance conditions to hold uniformly, rather than only along one favorable sequence of true parameters.

The corrected vector is generally dense. That is compatible with its role in inference: zeros in a sparse point estimate and approximately unbiased coordinate pivots serve different purposes. In finite samples, a remainder bound can still be numerically substantial even when an asymptotic condition will eventually hold. Report the model, variance convention, correction construction, and target coordinate before interpreting an interval. A normal-looking leading term does not excuse leaving its accompanying error term unchecked.