Consider a generalised linear model with parameter $\beta^{\top}$ partitioned as $\left(\beta_{0}^{\top}, \beta_{1}^{\top}\right)$, where $\beta_{0}$ has $p_{0}$ components and $\beta_{1}$ has $p-p_{0}$ components, and consider testing $H_{0}: \beta_{1}=0$ against $H_{1}: \beta_{1} \neq 0$. Define carefully the deviance, and use it to construct a test for $H_{0}$.

[You may use Wilks' theorem to justify this test, and you may also assume that the dispersion parameter is known.]

Now consider the generalised linear model with Poisson responses and the canonical link function with linear predictor $\eta=\left(\eta_{1}, \ldots, \eta_{n}\right)^{T}$ given by $\eta_{i}=x_{i}^{\top} \beta, i=1, \ldots, n$, where $x_{i 1}=1$ for every $i$. Derive the deviance for this model, and argue that it may be approximated by Pearson's $\chi^{2}$ statistic.