In the model $\left\{\mathcal{N}\left(\theta, I_{p}\right), \theta \in \mathbb{R}^{p}\right\}$ of a Gaussian distribution in dimension $p$, with unknown mean $\theta$ and known identity covariance matrix $I_{p}$, we estimate $\theta$ based on a sample of i.i.d. observations $X_{1}, \ldots, X_{n}$ drawn from $\mathcal{N}\left(\theta_{0}, I_{p}\right)$.

(a) Define the Fisher information $I\left(\theta_{0}\right)$, and compute it in this model.

(b) We recall that the observed Fisher information $i_{n}(\theta)$ is given by

$$i_{n}(\theta)=\frac{1}{n} \sum_{i=1}^{n} \nabla_{\theta} \log f\left(X_{i}, \theta\right) \nabla_{\theta} \log f\left(X_{i}, \theta\right)^{\top}$$

Find the limit of $\hat{i}_{n}=i_{n}\left(\hat{\theta}_{M L E}\right)$, where $\hat{\theta}_{M L E}$ is the maximum likelihood estimator of $\theta$ in this model.

(c) Define the Wald statistic $W_{n}(\theta)$ and compute it. Give the limiting distribution of $W_{n}\left(\theta_{0}\right)$ and explain how it can be used to design a confidence interval for $\theta_{0}$.

[You may use results from the course provided that you state them clearly.]