(a) Define a generalised linear model $(\mathrm{GLM})$ with design matrix $X \in \mathbb{R}^{n \times p}$, output variables $Y:=\left(Y_{1}, \ldots, Y_{n}\right)^{T}$ and parameters $\mu:=\left(\mu_{1}, \ldots, \mu_{n}\right)^{T}, \beta \in \mathbb{R}^{p}$ and $\sigma_{i}^{2}:=a_{i} \sigma^{2} \in(0, \infty), i=1, \ldots, n$. Derive the moment generating function of $Y$, i.e. give an expression for $\mathbb{E}\left[\exp \left(t^{T} Y\right)\right], t \in \mathbb{R}^{n}$, wherever it is well-defined.

Assume from now on that the GLM satisfies the usual regularity assumptions, $X$ has full column rank, and $\sigma^{2}$ is known and satisfies $1 / \sigma^{2} \in \mathbb{N}$.

(b) Let $\tilde{Y}:=\left(\tilde{Y}_{1}, \ldots, \tilde{Y}_{n / \sigma^{2}}\right)^{T}$ be the output variables of a GLM from the same family as that of part (a) and parameters $\tilde{\mu}:=\left(\tilde{\mu}_{1}, \ldots, \tilde{\mu}_{n / \sigma^{2}}\right)^{T}$ and $\tilde{\sigma}^{2}:=\left(\tilde{\sigma}_{1}^{2}, \ldots, \tilde{\sigma}_{n / \sigma^{2}}^{2}\right)$. Suppose the output variables may be split into $n$ blocks of size $1 / \sigma^{2}$ with constant parameters. To be precise, for each block $i=1, \ldots, n$, if $j \in\left\{(i-1) / \sigma^{2}+1, \ldots, i / \sigma^{2}\right\}$ then

$$\tilde{\mu}_{j}=\mu_{i} \quad \text { and } \quad \tilde{\sigma}_{j}^{2}=a_{i}$$

with $\mu_{i}=\mu_{i}(\beta)$ and $a_{i}$ defined as in part $($ a $)$. Let $\bar{Y}:=\left(\bar{Y}_{1}, \ldots, \bar{Y}_{n}\right)^{T}$, where $\bar{Y}_{i}:=$ $\sigma^{2} \sum_{k=1}^{1 / \sigma^{2}} \tilde{Y}_{(i-1) / \sigma^{2}+k}$.

(i) Show that $\bar{Y}$ is equal to $Y$ in distribution. [Hint: you may use without proof that moment generating functions uniquely determine distributions from exponential dispersion families.]

(ii) For any $\tilde{y} \in \mathbb{R}^{n / \sigma^{2}}$, let $\bar{y}=\left(\bar{y}_{1}, \ldots, \bar{y}_{n}\right)^{T}$, where $\bar{y}_{i}:=\sigma^{2} \sum_{k=1}^{1 / \sigma^{2}} \tilde{y}_{(i-1) / \sigma^{2}+k}$. Show that the model function of $\tilde{Y}$ satisfies

$$f\left(\tilde{y} ; \tilde{\mu}, \tilde{\sigma}^{2}\right)=g_{1}\left(\bar{y} ; \tilde{\mu}, \tilde{\sigma}^{2}\right) \times g_{2}\left(\tilde{y} ; \tilde{\sigma}^{2}\right)$$

for some functions $g_{1}, g_{2}$, and conclude that $\bar{Y}$ is a sufficient statistic for $\beta$ from $\tilde{Y}$.

(iii) For the model and data from part (a), let $\hat{\mu}$ be the maximum likelihood estimator for $\mu$ and let $D(Y ; \mu)$ be the deviance at $\mu$. Using (i) and (ii), show that

$$\frac{D(Y ; \hat{\mu})}{\sigma^{2}}={ }^{d} 2 \log \left\{\frac{\sup _{\tilde{\mu}^{\prime} \in \widetilde{\mathcal{M}}_{1}} f\left(\tilde{Y} ; \tilde{\mu}^{\prime}, \tilde{\sigma}^{2}\right)}{\sup _{\tilde{\mu}^{\prime} \in \widetilde{\mathcal{M}}_{0}} f\left(\tilde{Y} ; \tilde{\mu}^{\prime}, \tilde{\sigma}^{2}\right)}\right\}$$

where $=^{d}$ means equality in distribution and $\widetilde{\mathcal{M}}_{0}$ and $\widetilde{\mathcal{M}}_{1}$ are nested subspaces of $\mathbb{R}^{n / \sigma^{2}}$ which you should specify. Argue that $\operatorname{dim}\left(\widetilde{\mathcal{M}}_{1}\right)=n$ and $\operatorname{dim}\left(\widetilde{\mathcal{M}}_{0}\right)=p$, and, assuming the usual regularity assumptions, conclude that

$$\frac{D(Y ; \hat{\mu})}{\sigma^{2}} \rightarrow^{d} \chi_{n-p}^{2} \quad \text { as } \sigma^{2} \rightarrow 0$$

stating the name of the result from class that you use.