 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2020/ii/paper-4/13j/b/iii/solution

Codex  0 Created 2026-09-24 Updated 2026-09-29

Let

M_{1} = {(ν_{1} 1_{m}, \dots, ν_{n} 1_{m}) : (ν_{1}, \dots, ν_{n}) \in R^{n}}

be the block-saturated mean space, and let

M_{0} = {(μ_{1} (β) 1_{m}, \dots, μ_{n} (β) 1_{m}) : β \in R^{p}}

be the fitted GLM mean space. Full column rank and the regularity assumptions give dimensions

n

and

p

. The factorization in part (ii) makes the nuisance factor cancel from their likelihood ratio, while

\overset{ˉ}{Y} = d Y

gives

\frac{D ( Y ; μ )}{σ ^{2}} = d 2 lo g \frac{s u p _{\tilde{μ} \in M_{1}} f ( Y ; μ ~ , σ ~ ^{2} )}{s u p _{\tilde{μ} \in M_{0}} f ( Y ; μ ~ , σ ~ ^{2} )} .

σ^{2} \to 0

, the replicate count

m = 1/ σ^{2}

tends to infinity. Wilks theorem applied to these nested regular models makes twice the log-likelihood ratio converge in distribution to a chi-squared variable with the difference in dimensions:

\frac{D ( Y ; μ )}{σ ^{2}} d χ_{n - p}^{2} .

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