(\mathbf A ^{-1} \mathbf U) ( \mathbf U^T \mathbf A ^{-1}) } {1+ \mathbf U^T \mathbf A ^{-1}\mathbf U}.\end{aligned}$$ Moreover, we also know that $$\begin{aligned}
\label{det}
det(\mathbf A +\mathbf U \mathbf U^T) = det(\mathbf A) (1+\mathbf U^T \mathbf A^{-1} \mathbf U).\end{aligned}$$ Equation suggests that we can calculate the inverse of $(\mathbf A + \mathbf U \mathbf U^T)$ when we know the inverse of $\mathbf A$; Equation suggests a way of calculating the determinant of $\mathbf A +\mathbf U \mathbf U^T$ when we know both the determinant and inverse of $\mathbf A$. For the sequential estimation procedures discussed here, if we are at the $k$-th stage, then $\mathbf A = \left(\sum_{i=1}^k{\bm{{x}}_{ji}}{\bm{{x}}_{ji}^{\top}}\right)$ and $\mathbf U = {\bm{{x}}}_{j(k+1)}$. Because we need to repeatedly calculate the inverse and determinant of the design matrix when we recruit a new observation into the procedure at each stage, these two formulas are useful, as they save us a considerable amount of computation time.

### Approximation method {#approximation-method .unnumbered}

Let $\rho_j=N_j/N^*$, as before; then, with a probability of 1 $$\left(\sum_{j=1}^M\frac{\rho_j^2}{N_j}\right)^{-1}\left\{\sum_{j=1}^M \rho_j^2({\bm{{X}}_{jN_j}}{\bm{{X}}_{jN_j}^{\top}})^{-1} \right\}-N^*\left(\sum_{j=1}^M {\bm{{X}}_{jN_j}}{\bm{{X}}_{jN_j}^{\top}}\right)^{-1}\longrightarrow 0,$$ as $d$ goes to 0. Based on this fact, we define an alternative confidence set for $\beta_0$ as follows. $$\begin{aligned}
\label{FRn1}
R^{'}_{N^*}\!\! =\left\{Z\in R^p: 