({\hat{{\bm{{\beta}}}}}-{\bm{{z}}})^\top \left(\sum_{j=1}^M {\bm{{X}}_{jN_j}}{\bm{{X}}_{jN_j}^{\top}}\right)({\hat{{\bm{{\beta}}}}}-{\bm{{z}}})\leq \frac{N^*d^2}{\mu_{N^*}}\right\}.\end{aligned}$$ Obviously, it is easy to calculate when the number of dimensions $p$ is large, compared to (\[FRn\]). In addition, we show that the length of the maximum axis of the ellipsoid $R^{'}_{N^*}$ is no greater than $2d$, and that its coverage probability is approximately equal to the nominated $1-\alpha$ when the sample size is sufficiently large (or, equivalently, when $d$ is small) (see Appendix B). Hence, besides the aforementioned iterative formulas, these asymptotic results suggest that we can use $R^{'}_{N^*}$ as an alternative confidence set, and that they provide us another way of reducing the computation cost of calculating the inverse of the high-dimensional matrix.

On the other hand, we know that $$\begin{aligned}
    \left(\sum_{j=1}^M X_{jN_j}X_{jN_j}^{T}\right)\left\{\sum_{j=1}^M \rho_j^2(X_{jN_j}X_{jN_j}^{T})^{-1}\right\}
    &=\left(\sum_{j=1}^M \rho_j^2\right)I+\sum_{j\neq l}\rho_l^2 X_{jN_j}X_{jN_j}^{T}(X_{lN_l}X_{lN_l}^{T})^{-1} \nonumber\\
    &\geq \left(\sum_{j=1}^M \rho_j^2\right)I + \sum_{j\neq l}\rho_l \rho_j I=I,\label{I-ieq}\end{aligned}$$ where $I$ is an identical matrix with rank $p$. This implies that $$\begin{aligned}
    &\lambda_{max}\left\{\left(\sum_{j=1}^M X_{jN_j}X_{jN_j}^{T}\right)^{-1}\right\}\leq \lambda_{max}\left[\left\{\sum_{j=1}^M \rho_j^2(X_{jN_j}X_{jN_j}^{T})^{-1}\right\}\right].\label{comp.eigen}\end{aligned}$$ To have the same coverage probability, the confidence set based on (\[FRn1\]) requires more observations. Therefore, we suggest that users use a confidence set based on (\[FRn1\]) only for models with a lengthy parameter list, and/or when the cost of computation is high. Otherwise, we recommend using the confidence region 