Let $X_{1}, \ldots, X_{n} \sim$ iid $\operatorname{Gamma}(\alpha, \beta)$ for some known $\alpha>0$ and some unknown $\beta>0$. [The gamma distribution has probability density function

$$f(x)=\frac{\beta^{\alpha}}{\Gamma(\alpha)} x^{\alpha-1} e^{-\beta x}, \quad x>0$$

and its mean and variance are $\alpha / \beta$ and $\alpha / \beta^{2}$, respectively.]

(a) Find the maximum likelihood estimator $\hat{\beta}$ for $\beta$ and derive the distributional limit of $\sqrt{n}(\hat{\beta}-\beta)$. [You may not use the asymptotic normality of the maximum likelihood estimator proved in the course.]

(b) Construct an asymptotic $(1-\gamma)$-level confidence interval for $\beta$ and show that it has the correct (asymptotic) coverage.

(c) Write down all the steps needed to construct a candidate to an asymptotic $(1-\gamma)$-level confidence interval for $\beta$ using the nonparametric bootstrap.