Consider the linear model $Y=X \beta+\epsilon$ where $Y=\left(Y_{1}, \ldots, Y_{n}\right)^{\mathrm{T}}, \beta=\left(\beta_{1}, \ldots, \beta_{p}\right)^{\mathrm{T}}$, and $\epsilon=\left(\epsilon_{1}, \ldots, \epsilon_{n}\right)^{\mathrm{T}}$, with $\epsilon_{1}, \ldots, \epsilon_{n}$ independent $N\left(0, \sigma^{2}\right)$ random variables. The $(n \times p)$ matrix $X$ is known and is of full rank $p<n$. Give expressions for the maximum likelihood estimators $\widehat{\beta}$ and $\widehat{\sigma}^{2}$ of $\beta$ and $\sigma^{2}$ respectively, and state their joint distribution. Show that $\widehat{\beta}$ is unbiased whereas $\widehat{\sigma}^{2}$ is biased.

Suppose that a new variable $Y^{*}$ is to be observed, satisfying the relationship

$$Y^{*}=x^{* \mathrm{~T}} \beta+\epsilon^{*}$$

where $x^{*}(p \times 1)$ is known, and $\epsilon^{*} \sim N\left(0, \sigma^{2}\right)$ independently of $\epsilon$. We propose to predict $Y^{*}$ by $\widetilde{Y}=x^{* \mathrm{~T}} \widehat{\beta}$. Identify the distribution of

$$\frac{Y^{*}-\tilde{Y}}{\tau \tilde{\sigma}}$$

where

$$\begin{aligned} \tilde{\sigma}^{2} &=\frac{n}{n-p} \widehat{\sigma}^{2} \\ \tau^{2} &=x^{* \mathrm{~T}}\left(X^{\mathrm{T}} X\right)^{-1} x^{*}+1 \end{aligned}$$