Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 1 / 5K (Statistical Modelling)

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- Solution 5K

Solution

The normal linear model is

Y = Xβ + ε

with

ε \sim N (0, σ^{2} I_{n})

. Full column rank gives

β = (X^{T} X)^{- 1} X^{T} Y .

With 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) ε

and

I - H

is an orthogonal projection,

Var (e_{i}) = σ^{2} (1 - h_{ii}) .

High-leverage observations can move their own fitted values strongly and may exert disproportionate influence; residual size alone can therefore conceal them.

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