(a) Consider the general linear model $Y=X \theta+\varepsilon$ where $X$ is a known $n \times p$ matrix, $\theta$ is an unknown $p \times 1$ vector of parameters, and $\varepsilon$ is an $n \times 1$ vector of independent $N\left(0, \sigma^{2}\right)$ random variables with unknown variances $\sigma^{2}$. Show that, provided the matrix $X$ is of rank $p$, the least squares estimate of $\theta$ is

$$\hat{\theta}=\left(X^{\mathrm{T}} X\right)^{-1} X^{\mathrm{T}} Y$$

Let

$$\hat{\varepsilon}=Y-X \hat{\theta}$$

What is the distribution of $\hat{\varepsilon}^{\mathrm{T}} \hat{\varepsilon}$ ? Write down, in terms of $\hat{\varepsilon}^{\mathrm{T}} \hat{\varepsilon}$, an unbiased estimator of $\sigma^{2}$.

(b) Four points on the ground form the vertices of a plane quadrilateral with interior angles $\theta_{1}, \theta_{2}, \theta_{3}, \theta_{4}$, so that $\theta_{1}+\theta_{2}+\theta_{3}+\theta_{4}=2 \pi$. Aerial observations $Z_{1}, Z_{2}, Z_{3}, Z_{4}$ are made of these angles, where the observations are subject to independent errors distributed as $N\left(0, \sigma^{2}\right)$ random variables.

(i) Represent the preceding model as a general linear model with observations $\left(Z_{1}, Z_{2}, Z_{3}, Z_{4}-2 \pi\right)$ and unknown parameters $\left(\theta_{1}, \theta_{2}, \theta_{3}\right)$.

(ii) Find the least squares estimates $\hat{\theta}_{1}, \hat{\theta}_{2}, \hat{\theta}_{3}$.

(iii) Determine an unbiased estimator of $\sigma^{2}$. What is its distribution?