(a) State and prove the Cramér-Rao inequality in a parametric model $\{f(\theta): \theta \in \Theta\}$, where $\Theta \subseteq \mathbb{R}$. [Necessary regularity conditions on the model need not be specified.]

(b) Let $X_{1}, \ldots, X_{n}$ be i.i.d. Poisson random variables with unknown parameter $E X_{1}=\theta>0$. For $\bar{X}_{n}=(1 / n) \sum_{i=1}^{n} X_{i}$ and $S^{2}=(n-1)^{-1} \sum_{i=1}^{n}\left(X_{i}-\bar{X}_{n}\right)^{2}$ define

$$T_{\alpha}=\alpha \bar{X}_{n}+(1-\alpha) S^{2}, \quad 0 \leqslant \alpha \leqslant 1$$

Show that $\operatorname{Var}_{\theta}\left(T_{\alpha}\right) \geqslant \operatorname{Var}_{\theta}\left(\bar{X}_{n}\right)$ for all values of $\alpha, \theta$.

Now suppose $\tilde{\theta}=\tilde{\theta}\left(X_{1}, \ldots, X_{n}\right)$ is an estimator of $\theta$ with possibly nonzero bias $B(\theta)=E_{\theta} \tilde{\theta}-\theta$. Suppose the function $B$ is monotone increasing on $(0, \infty)$. Prove that the mean-squared errors satisfy

$$E_{\theta}\left(\tilde{\theta}_{n}-\theta\right)^{2} \geqslant E_{\theta}\left(\bar{X}_{n}-\theta\right)^{2} \text { for all } \theta \in \Theta$$