Show that the least-squares estimate of \(\boldsymbol{\beta}\) (say \(\hat{\boldsymbol{\beta}}_{(i)}\) ) with the \(i\) th observation deleted can be written in terms of the estimate based on all \(n\) points as

\[
\hat{\beta}_{(i)}=\hat{\beta}-\frac{e_{i}}{1-h_{i i}}\left(\mathbf{X}^{\prime} \mathbf{X}\right)^{-1} \mathbf{x}_{i}
\]