each procedure with random weights $\rho_j=\tilde{N}_j/\tilde{N}$, $j=1,\ldots, M$. We know that for each $j$, $$\begin{aligned}
    \sqrt{\tilde{N}_j}({\bm{{\beta}}_{j}^{*}}-{\bm{{\beta}}_{0}})\longrightarrow N(0,\sigma^{2}{{\bm{{I}}_{0}}}\Sigma^{-1} {{\bm{{I}}_{0}}})
~~\text{\rm in distribution as }d\rightarrow 0,\nonumber\end{aligned}$$ where ${\bm{{I}}_{0}}=\text{diag}\{I(\beta_{01}\neq 0),...,I(\beta_{0p}\neq 0)\}$. Because each observation will be recruited into one procedure only, ${\bm{{\beta}}_{j}^{*}}, j=1,...,M$, are independent. This implies that $$\begin{aligned}
\label{Tasynorm}
    \sqrt{\tilde{N}}({\tilde{{\bm{{\beta}}}}}-{\bm{{\beta}}_{0}})\longrightarrow N(0,\sigma^{2}{{\bm{{I}}_{0}}}\Sigma^{-1} {{\bm{{I}}_{0}}})
~~\text{\rm in distribution as }d\rightarrow 0.\end{aligned}$$ Hence, if $\beta_{0l}=0$, for some $1 \leq l \leq p$, $\sqrt{\tilde{N}}(\tilde{\beta}_{l}-\beta_{0l})$ will degenerate to the constant 0 when the sample size is large ($d$ tends to 0); otherwise, it retains the same distribution as that of its corresponding component in the LSE. Thus, Equation (\[Tasynorm\]) implies that as $d\rightarrow 0$, $$\begin{aligned}
    \hat\sigma^{-2}({\tilde{{\bm{{\beta}}}}}-{\bm{{\beta}}_{0}})^\top \left[{\bm{{I}}^{*}}\left\{\sum_{j=1}^M {\bm{{X}}_{jN_j}}{\bm{{X}}_{jN_j}^{\top}}\right\}^{-1}{\bm{{I}}^{*}}\right]^{-}({\tilde{{\bm{{\beta}}}}}-{\bm{{\beta}}_{0}})\longrightarrow \chi_{p_0}^2,\label{dist-var}\end{aligned}$$ where notation ${\bm{{A}}^{-}}$ denotes a general inverse of matrix ${\bm{{A}}}$.

Let ${\bm{{O}}}$ be an orthonormal matrix, depending on the current ASE estimate, such that $({\tilde{{\bm{{\beta}}}}_{1}},{\tilde{{\bm{{\beta}}}}_{2}})^{\top}={\bm{{O}}}{\tilde{{\bm{{\beta}}}}}$ and ${\bm{{O}}}{\bm{{O}}^{\top}}=I_p$, where ${\tilde{{\bm{{\beta}}}}_{2}}=0$ and ${\tilde{{\bm{{\beta}}}}_{1}}$ includes all nonzero elements of ${\tilde{{\bm{{\beta}}}}}$. We show in Appendix D that $$\begin{aligned}
    &\hat\sigma^{-2}({\tilde{{\bm{{\beta}}}}}-{\bm{{\beta}}_{0}})^\top \left[{\bm{{I}}^{*}}\left\{\sum_{j=1}^M {\bm{{X}}_{jN_j}}{\bm{{X}}_{jN_j}^{\top}}\right\}^{-1}{\bm{{I}}^{*}}\right]^{-}({\tilde{{\bm{{\beta}}}}}-{\bm{{\beta}}_{0}})\nonumber\\
    & =\hat\sigma^{-2}({\tilde{{\bm{{\beta}}}}_{1}}-{\bm{{\beta}}_{01}})^\top {\tilde{{\bm{{\Sigma}}}}_{11}}({\tilde{{\bm{{\beta}}}}_{1}}-{\bm{{\beta}}_{01}}),\end{aligned}$$ where ${\tilde{{\bm{{\Sigma}}}}_{11}}$ is defined as (\[matrix\]) in Appendix D. Based on this, we define a confidence set of ${\tilde{{\bm{{\beta}}}}}$ as follows. $$\begin{aligned}
\label{Rn-var}
    R_{\tilde {N}}\!\! =\!\left\{ {\bm{{Z}}} \in R^p\!\!:~~\!\!\frac{S_{\tilde {N}}}{\tilde {N}}\leq \frac{d^2}{\nu_{\tilde {N}}}\! \mbox{ and }\! z_j\!=0 \!\mbox{ for 