ance matrix estimation, it is all that is needed to apply the Gaussian comparison theorem \[thm:comparisons\], which will allow us to extend the Berry-Esseen bound established in to the case when $\Gamma$ is estimated.

\[lemma::upsilon\] Let $$\label{eq:aleph}
\aleph_n = \max \Big\{ \overline{H} B \overline{v} \sqrt{ b\frac{   
 \log n}{n}},  B^2 
 \sqrt{   b \overline{v}  \frac{\log b + \log n }{n}   }\Big\}.$$ There exists a $C > 0$ dependent on $A$ only such that $$\label{eq:upsilon}
\sup_{P\in {\cal P}_n}
\mathbb{P}\left(\max_{j,l} \left| \hat\Gamma(j,l)-\Gamma(j,l)\right| \geq C
\,\aleph_n\right) \leq \frac{2}{n}.$$

Now we construct the confidence set. Let $Q=(Q(1),\ldots, Q(s))$ be i.i.d. standard Normal variables, independent of the data. Let $\hat Z = \hat\Gamma^{1/2} Q$ and define $\hat{t}_\alpha$ by $$\label{eq:hat.t.alpha.berry}
\mathbb{P}( ||\hat Z||_\infty > \hat{t}_\alpha \,|\, \hat\Gamma)=\alpha.$$ Finally, let $$\label{eq::conf-rectangle}
    \hat{C}_n = \Bigl\{ \theta \in \mathbb{R}^s:\ ||\theta-\hat\theta||_\infty \leq \frac{\hat{t}_\alpha}{\sqrt{n}}\Bigr\}.$$

\[thm::coverage\] There exists a $C>0$, dependent only on $A$, such that $$\inf_{P\in {\cal P}}\mathbb{P}(\theta\in R_n)= 1-\alpha -
C \left( \Delta_{n,1} + \Delta_{n,2} + \Delta_{n,3} + \frac{1}{n} \right),$$ where $$\label{eq::this-is-upsilon}
\Delta_{n,3}=  \frac{\aleph_n^{1/3} (2 \log 2s)^{2/3}}{\underline{\sigma}^{2/3}}.$$

[**Remark**]{}. The additional term $\Delta_{n,3}$ in the previous theorem is due to the uncertainty in estimating $\Gamma$, and can be established by using the comparison inequality for Gaussian vectors of [@chernozhukov2015comparison], keeping track of the dependence on $\underline{\sigma}^2$; see below.

In addition to $L_\infty$ balls, we can also construct our confidence set to be a hyper-rectangle, with side lengths proportional to the standard errors of the projection parameters. That is, we define $$\label{eq:hyper:CI}
\tilde C_n = \bigotimes_{j\in S} C(j),$$ where $$C(j) = \left[    \hat\beta_S(j) - z_{