Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 1 / 13J / iv / Solution

Let ${\hat{\beta }}_{(i)}$ be the least-squares estimate after deleting observation $i$, and put $\hat{Y}=X\hat{\beta }$ and ${\hat{Y}}_{(i)}=X{\hat{\beta }}_{(i)}$. Cook's distance isThus $D_i$ is the squared displacement caused by deleting observation $i$, measured in the same $X^{T}X$ metric and scale as the confidence ellipsoid. In particular, ${\hat{\beta }}_{(i)}$ lies outside the $(1-\alpha )$ ellipsoid centered at $\hat{\beta }$ exactly when