For the statistical model $\left\{\mathcal{N}_{d}(\theta, \Sigma), \theta \in \mathbb{R}^{d}\right\}$, where $\Sigma$ is a known, positive-definite $d \times d$ matrix, we want to estimate $\theta$ based on $n$ i.i.d. observations $X_{1}, \ldots, X_{n}$ with distribution $\mathcal{N}_{d}(\theta, \Sigma)$.

(a) Derive the maximum likelihood estimator $\hat{\theta}_{n}$ of $\theta$. What is the distribution of $\hat{\theta}_{n}$ ?

(b) For $\alpha \in(0,1)$, construct a confidence region $C_{n}^{\alpha}$ such that $\mathbf{P}_{\theta}\left(\theta \in C_{n}^{\alpha}\right)=1-\alpha$.

(c) For $\Sigma=I_{d}$, compute the maximum likelihood estimator of $\theta$ for the following parameter spaces:

(i) $\Theta=\left\{\theta:\|\theta\|_{2}=1\right\}$.

(ii) $\Theta=\left\{\theta: v^{\top} \theta=0\right\}$ for some unit vector $v \in \mathbb{R}^{d}$.

(d) For $\Sigma=I_{d}$, we want to test the null hypothesis $\Theta_{0}=\{0\}$ (i.e. $\left.\theta=0\right)$ against the composite alternative $\Theta_{1}=\mathbb{R}^{d} \backslash\{0\}$. Compute the likelihood ratio statistic $\Lambda\left(\Theta_{1}, \Theta_{0}\right)$ and give its distribution under the null hypothesis. Compare this result with the statement of Wilks' theorem.