Let $S_{N_j}=({\bm{{z}}}-{\hat{{\bm{{\beta}}}}_{jN_j}})^{\top}({\bm{{X}}_{jN_j}}{\bm{{X}}_{jN_j}^{\top}}) ({\bm{{z}}}-{\hat{{\bm{{\beta}}}}_{jN_j}})$, and ${\bm{{z}}}=(z_1,\cdots, z_p)^{\top}$ and $$\begin{aligned}
\label{Rn}
    R_{N_j}\!\! =\!\left\{ {\bm{{z}}} \in R^p\!\!:~~\!\!\frac{S_{N_j}}{N_j}\leq \frac{d^2}{\mu_{jN_j}}\right\}.\end{aligned}$$ When the $j$th procedure is stopped according to , it is shown that $R_{N_j}$ is a confidence ellipsoid of ${\bm{{\beta}}_{0}}$ with the length of a maximum axis no greater than $2d$ and the coverage probability equals approximately $1-\tilde{\alpha}_j$, where $\tilde{\alpha}_j$ is the probability corresponding to $\tilde a_j$, [which is the $1-\tilde{\alpha}_j$ quantile of $\chi^2(p)$]{}. The coverage probability of $R_{N_j}$ depends on both $\tilde a_j$ and $d$. The constant $d>0$ serves as a precision index, which restricts the size of the confidence ellipsoid; this can be determined by users, based on their practical needs. When $d$ becomes smaller, we will need more observations, such that the estimate can achieve the desired precision. There have been many discussions in the literature regarding this conventional sequential estimation procedure; hence we only briefly touch upon technical matters, in Appendix A.

[ In a conventional sequential confidence set estimation procedure (i.e. $M=1$), we first specify the coverage, and then use its corresponding quantile and the size of the confidence set to define the stopping time. In the current method, we do not specify the coverage probability for each procedure $j$, 