We consider the problem of estimating an unknown $\theta_{0}$ in a statistical model $\{f(x, \theta), \theta \in \Theta\}$ where $\Theta \subset \mathbb{R}$, based on $n$ i.i.d. observations $X_{1}, \ldots, X_{n}$ whose distribution has p.d.f. $f\left(x, \theta_{0}\right)$.

In all the parts below you may assume that the model satisfies necessary regularity conditions.

(a) Define the score function $S_{n}$ of $\theta$. Prove that $S_{n}\left(\theta_{0}\right)$ has mean 0 .

(b) Define the Fisher Information $I(\theta)$. Show that it can also be expressed as

$$I(\theta)=-\mathbb{E}_{\theta}\left[\frac{d^{2}}{d \theta^{2}} \log f\left(X_{1}, \theta\right)\right]$$

(c) Define the maximum likelihood estimator $\hat{\theta}_{n}$ of $\theta$. Give without proof the limits of $\hat{\theta}_{n}$ and of $\sqrt{n}\left(\hat{\theta}_{n}-\theta_{0}\right.$ ) (in a manner which you should specify). [Be as precise as possible when describing a distribution.]

(d) Let $\psi: \Theta \rightarrow \mathbb{R}$ be a continuously differentiable function, and $\tilde{\theta}_{n}$ another estimator of $\theta_{0}$ such that $\left|\hat{\theta}_{n}-\tilde{\theta}_{n}\right| \leqslant 1 / n$ with probability 1 . Give the limits of $\psi\left(\tilde{\theta}_{n}\right)$ and of $\sqrt{n}\left(\psi\left(\tilde{\theta}_{n}\right)-\psi\left(\theta_{0}\right)\right)$ (in a manner which you should specify).