(i) Suppose that $Y_{1}, \ldots, Y_{n}$ are independent random variables, and that $Y_{i}$ has probability density function

$$f\left(y_{i} \mid \beta, \nu\right)=\left(\frac{\nu y_{i}}{\mu_{i}}\right)^{\nu} e^{-y_{i} \nu / \mu_{i}} \frac{1}{\Gamma(\nu)} \frac{1}{y_{i}} \text { for } y_{i}>0$$

where

$$1 / \mu_{i}=\beta^{T} x_{i}, \text { for } \quad 1 \leqslant i \leqslant n,$$

and $x_{1}, \ldots, x_{n}$ are given $p$-dimensional vectors, and $\nu$ is known.

Show that $\mathbb{E}\left(Y_{i}\right)=\mu_{i}$ and that $\operatorname{var}\left(Y_{i}\right)=\mu_{i}^{2} / \nu$.

(ii) Find the equation for $\hat{\beta}$, the maximum likelihood estimator of $\beta$, and suggest an iterative scheme for its solution.

(iii) If $p=2$, and $x_{i}=\left(\begin{array}{c}1 \\ z_{i}\end{array}\right)$, find the large-sample distribution of $\hat{\beta}_{2}$. Write your answer in terms of $a, b, c$ and $\nu$, where $a, b, c$ are defined by

$$a=\sum \mu_{i}^{2}, \quad b=\sum z_{i} \mu_{i}^{2}, \quad c=\sum z_{i}^{2} \mu_{i}^{2} .$$