n_0~~\mbox{and}~~(\hat\sigma_{jk}^2+\frac{1}{k})\leq \frac{d^2k}{\tilde{a}_{jk}^2\tilde{\mu}_{jk}}\right\},\end{aligned}$$ where $\tilde{\mu}_{jk}=\lambda_{max}[k{\bm{{I}}_{j}}(\epsilon)({\bm{{X}}_{jk}}{\bm{{X}}_{jk}^{\top}})^{-1}{\bm{{I}}_{j}}(\epsilon)]$, and ${\bm{{X}}_{jk}}$ is the design matrix under the current stage.

We conduct procedure $j$ as before, until the stopping criterion $\tilde{N}_j$ is satisfied. Please note that $\tilde{a}_{jk}^2$ in (\[Ndef-var\]) is now random and depends on the estimated number of effective variables at the current stage. Because ${a}_{k}^2$ in (\[Ndef\]) is a constant calculated with a total number ($p$) of variables, this implies that $\tilde{a}_{jk}^2\leq {a}_{k}^2$ almost surely. Therefore, ${N}_j$ is stochastically larger than $\tilde{N}_j$; this is especially the case when $p_0$ is much smaller than $p$. These phenomena will also appear in our numerical studies.

Adaptive shrinkage estimation in distributed sequential estimation
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When we adopt the ASE in all procedures, the major concern will be whether or not the selected variables from each procedure will be consistent.

For each $j=1, \ldots, M$, let $\tilde{N}_j$, ${\bm{{\beta}}_{j}^{*}}$, and ${\bm{{I}}_{j}}(\epsilon)$ be defined as before. Let $\tilde{N}=\sum_{j=1}^M \tilde{N}_j$ be the sum of the stopping time: when all procedures are stopped, it is equal to the sum of observations used in each procedure. Let $${\bm{{I}}^{*}}=\prod_{j=1}^M {\bm{{I}}_{j}}(\epsilon)$$ be an element-wise product of the indicator vectors and define the combined ASE $${\tilde{{\bm{{\beta}}}}}={\bm{{I}}^{*}}(\sum_{j=1}^M \rho_j{\bm{{\beta}}_{j}^{*}}),$$ as a weighted average of the ASE ${\bm{{\beta}}_{j}^{*}}$ of 