$p_0$ of $p$ components of ${\bm{{\mathbf\beta}}_{0}}$ are effective to the model, and that we want to simultaneously identify these variables and construct a fixed-size confidence set with a prescribed coverage probability for them. Let ${\hat{{\bm{{\beta}}}}_{j}}$ be the LSE of ${\bm{{\beta}}_{0}}$ in procedure $j$, as before, and let $\lambda_k=\lambda |\hat{\beta}_{j}^k|^{-\gamma}$, where $\hat{\beta}_{j}^k$ is the $k$th element of ${\hat{{\bm{{\beta}}}}_{j}}$. In addition, let $\lambda\equiv\lambda(n)$ be a nonrandom function of $n$, such that for some $ 0<\delta<1/2 $ and $\gamma >0$, $$\begin{aligned}
\label{cond1}
n^{1/2}\lambda \rightarrow 0 \text{ and }
n^{1/2+\gamma\delta}\lambda\longrightarrow \infty, \mbox{ as }
n\rightarrow \infty.\end{aligned}$$ Let $\epsilon >0 $ be constant; then, define $I_{j}^k(\epsilon)=I(\sqrt n \lambda_k<\epsilon)$, for $k =1, \ldots, p$ as an indicator function, and let $ {\bm{{I}}_{j}}(\epsilon) =\text{diag}\{I_{j}^1(\epsilon),\cdots,I_{j}^p(\epsilon)\}$ be a $p\times p$ diagonal matrix. Then, ${\bm{{\beta}}_{j}^{*}}\equiv{\bm{{I}}_{j}}(\epsilon){\hat{{\bm{{\beta}}}}_{j}}$ is an ASE of $\mathbf \beta_0$, where ${\beta}_{j}^{*k}=0$ if $I_{j}^k(\epsilon)=0$; otherwise, it remains the same as $\hat{\beta}_{j}^{k}$.

Suppose that $\hat\sigma_{jk}^2$ and $\hat p_0(k)=\sum_{l=1}^p I_{jl}(\epsilon)$ are estimates of $\sigma_{jk}^2$ and $p_0$, respectively, based on the observations in $C_{jk}$ at the $k$th stage. Let $\chi_{\hat p_0(k)}(\alpha)$ be an $1-\alpha$ quantile of the chi-square distribution with $\hat p_0(k)$ degrees of freedom, and let $\tilde{a}_{jk}^2=\chi_{\hat p_0(k)}(\alpha)/M$. Then, define a stopping rule for procedure $j$ as follows: $$\begin{aligned}
\label{Ndef-var}
    \tilde{N}_j=N_d\equiv\inf\left\{k:~k\geq 