Let $Y_{1}, \ldots, Y_{n}$ be independent with $Y_{i} \sim \frac{1}{n_{i}} \operatorname{Bin}\left(n_{i}, \mu_{i}\right), i=1, \ldots, n$, and

$$\log \left(\frac{\mu_{i}}{1-\mu_{i}}\right)=x_{i}^{\top} \beta$$

where $x_{i}$ is a $p \times 1$ vector of regressors and $\beta$ is a $p \times 1$ vector of parameters. Write down the likelihood of the data $Y_{1}, \ldots, Y_{n}$ as a function of $\mu=\left(\mu_{1}, \ldots, \mu_{n}\right)$. Find the unrestricted maximum likelihood estimator of $\mu$, and the form of the maximum likelihood estimator $\hat{\mu}=\left(\hat{\mu}_{1}, \ldots, \hat{\mu}_{n}\right)$ under the logistic model (1).

Show that the deviance for a comparison of the full (saturated) model to the generalised linear model with canonical link (1) using the maximum likelihood estimator $\hat{\beta}$ can be simplified to

$$D(y ; \hat{\mu})=-2 \sum_{i=1}^{n}\left[n_{i} y_{i} x_{i}^{\top} \hat{\beta}-n_{i} \log \left(1-\hat{\mu}_{i}\right)\right]$$

Finally, obtain an expression for the deviance residual in this generalised linear model.