= Solution
Let
$$
\widetilde{\mathcal M}_1
=\{(\nu_1\mathbf1_m,\ldots,\nu_n\mathbf1_m):
(\nu_1,\ldots,\nu_n)\in\mathbb R^n\}
$$
be the block-saturated mean space, and let
$$
\widetilde{\mathcal M}_0
=\{(\mu_1(\beta)\mathbf1_m,\ldots,
\mu_n(\beta)\mathbf1_m):\beta\in\mathbb R^p\}
$$
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 $\bar Y\overset d=Y$ gives
$$
\frac{D(Y;\widehat\mu)}{\sigma^2}
\overset d=
2\log
\frac{
\sup_{\tilde\mu\in\widetilde{\mathcal M}_1}
f(\widetilde Y;\tilde\mu,\tilde\sigma^2)}{
\sup_{\tilde\mu\in\widetilde{\mathcal M}_0}
f(\widetilde Y;\tilde\mu,\tilde\sigma^2)}.
$$
As $\sigma^2\to0$, 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:
$$
\boxed{
\frac{D(Y;\widehat\mu)}{\sigma^2}
\xrightarrow d\chi^2_{n-p}}.
$$
Back to article page
