(i) Let $X_{1}, \ldots, X_{n}$ be independent, identically-distributed $N\left(\mu, \mu^{2}\right)$ random variables, $\mu>0$.

Find a minimal sufficient statistic for $\mu$.

Let $T_{1}=n^{-1} \sum_{i=1}^{n} X_{i}$ and $T_{2}=\sqrt{n^{-1} \sum_{i=1}^{n} X_{i}^{2}}$. Write down the distribution of $X_{i} / \mu$, and hence show that $Z=T_{1} / T_{2}$ is ancillary. Explain briefly why the Conditionality Principle would lead to inference about $\mu$ being drawn from the conditional distribution of $T_{2}$ given $Z$.

What is the maximum likelihood estimator of $\mu$ ?

(ii) Describe briefly the Bayesian approach to predictive inference,

Let $Z_{1}, \ldots, Z_{n}$ be independent, identically-distributed $N\left(\mu, \sigma^{2}\right)$ random variables, with $\mu, \sigma^{2}$ both unknown. Derive the maximum likelihood estimators $\widehat{\mu}, \widehat{\sigma}^{2}$ of $\mu, \sigma^{2}$ based on $Z_{1}, \ldots, Z_{n}$, and state, without proof, their joint distribution.

Suppose that it is required to construct a prediction interval

$I_{1-\alpha} \equiv I_{1-\alpha}\left(Z_{1}, \ldots, Z_{n}\right)$ for a future, independent, random variable $Z_{0}$ with the same $N\left(\mu, \sigma^{2}\right)$ distribution, such that

$$P\left(Z_{0} \in I_{1-\alpha}\right)=1-\alpha$$

with the probability over the joint distribution of $Z_{0}, Z_{1}, \ldots, Z_{n}$. Let

$$I_{1-\alpha}\left(Z_{1}, \ldots, Z_{n} ; \sigma^{2}\right)=\left[\bar{Z}_{n}-z_{\alpha / 2} \sigma \sqrt{1+1 / n}, \bar{Z}_{n}+z_{\alpha / 2} \sigma \sqrt{1+1 / n}\right]$$

where $\bar{Z}_{n}=n^{-1} \sum_{i=1}^{n} Z_{i}$, and $\Phi\left(z_{\beta}\right)=1-\beta$, with $\Phi$ the distribution function of $N(0,1)$.

Show that $P\left(Z_{0} \in I_{1-\alpha}\left(Z_{1}, \ldots, Z_{n} ; \sigma^{2}\right)\right)=1-\alpha$.

By considering the distribution of $\left(Z_{0}-\bar{Z}_{n}\right) /\left(\widehat{\sigma} \sqrt{\frac{n+1}{n-1}}\right)$, or otherwise, show that

$$P\left(Z_{0} \in I_{1-\alpha}\left(Z_{1}, \ldots, Z_{n} ; \widehat{\sigma}^{2}\right)\right)<1-\alpha,$$

and show how to construct an interval $I_{1-\gamma}\left(Z_{1}, \ldots, Z_{n} ; \widehat{\sigma}^{2}\right)$ with

$$P\left(Z_{0} \in I_{1-\gamma}\left(Z_{1}, \ldots, Z_{n} ; \widehat{\sigma}^{2}\right)\right)=1-\alpha .$$

[Hint: if $Y$ has the $t$-distribution with $m$ degrees of freedom and $t_{\beta}^{(m)}$ is defined by $P\left(Y<t_{\beta}^{(m)}\right)=1-\beta$ then $t_{\beta}>z_{\beta}$ for $\beta<\frac{1}{2}$.]