Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 1 / 5K / Solution

The normal linear model is $Y=X\beta +\epsilon$ with $\epsilon \sim N(0,{\sigma }^2{I}_n)$. Full column rank givesWith hat matrix $H=X(X^{T}X{)}^{-1}{X}^T$, the $i$th regression leverage is $h_{ii}$. Since $e=(I-H)Y=(I-H)\epsilon$ and $I-H$ is an orthogonal projection,High-leverage observations can move their own fitted values strongly and may exert disproportionate influence; residual size alone can therefore conceal them.