Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 4 / 13J / b / iii / Solution

Letbe the block-saturated mean space, and letbe 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 $\bar{Y}\stackrel{d}{=}Y$ givesAs ${\sigma }^2\to 0$, the replicate count $m=1/{\sigma }^2$ 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: