We consider the exponential model $\{f(\cdot, \theta): \theta \in(0, \infty)\}$, where

$$f(x, \theta)=\theta e^{-\theta x} \quad \text { for } x \geqslant 0$$

We observe an i.i.d. sample $X_{1}, \ldots, X_{n}$ from the model.

(a) Compute the maximum likelihood estimator $\hat{\theta}_{M L E}$ for $\theta$. What is the limit in distribution of $\sqrt{n}\left(\hat{\theta}_{M L E}-\theta\right)$ ?

(b) Consider the Bayesian setting and place a $\operatorname{Gamma}(\alpha, \beta), \alpha, \beta>0$, prior for $\theta$ with density

$$\pi(\theta)=\frac{\beta^{\alpha}}{\Gamma(\alpha)} \theta^{\alpha-1} \exp (-\beta \theta) \quad \text { for } \theta>0$$

where $\Gamma$ is the Gamma function satisfying $\Gamma(\alpha+1)=\alpha \Gamma(\alpha)$ for all $\alpha>0$. What is the posterior distribution for $\theta$ ? What is the Bayes estimator $\hat{\theta}_{\pi}$ for the squared loss?

(c) Show that the Bayes estimator is consistent. What is the limiting distribution of $\sqrt{n}\left(\hat{\theta}_{\pi}-\theta\right)$ ?

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