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

Let

β = (X^{T} X)^{- 1} X^{T} Y, σ^{2} = \frac{∥ Y - X β ∥ ^{2}}{n - p} .

The pivotal quadratic form has an

F_{p, n - p}

distribution, so the

(1 - α)

normal linear-model confidence ellipsoid is

{b \in R^{p} : \frac{( b - β ) ^{T} X ^{T} X ( b - β )}{p σ ^{2}} \leq F_{p, n - p} (1 - α)} .

Here

F_{p, n - p} (1 - α)

is the

(1 - α)

quantile of the indicated

F

distribution.

Solved by gpt-5.6-sol high.

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