We consider the problem of estimating $\theta$ in the model $\{f(x, \theta): \theta \in(0, \infty)\}$, where

$$f(x, \theta)=(1-\alpha)(x-\theta)^{-\alpha} 1\{x \in[\theta, \theta+1]\}$$

Here $1\{A\}$ is the indicator of the set $A$, and $\alpha \in(0,1)$ is known. This estimation is based on a sample of $n$ i.i.d. $X_{1}, \ldots, X_{n}$, and we denote by $X_{(1)}<\ldots<X_{(n)}$ the ordered sample.

(a) Compute the mean and the variance of $X_{1}$. Construct an unbiased estimator of $\theta$ taking the form $\tilde{\theta}_{n}=\bar{X}_{n}+c(\alpha)$, where $\bar{X}_{n}=n^{-1} \sum_{i=1}^{n} X_{i}$, specifying $c(\alpha)$.

(b) Show that $\tilde{\theta}_{n}$ is consistent and find the limit in distribution of $\sqrt{n}\left(\tilde{\theta}_{n}-\theta\right)$. Justify your answer, citing theorems that you use.

(c) Find the maximum likelihood estimator $\hat{\theta}_{n}$ of $\theta$. Compute $\mathbf{P}\left(\hat{\theta}_{n}-\theta>t\right)$ for all real $t$. Is $\hat{\theta}_{n}$ unbiased?

(d) For $t>0$, show that $\mathbf{P}\left(n^{\beta}\left(\hat{\theta}_{n}-\theta\right)>t\right)$ has a limit in $(0,1)$ for some $\beta>0$. Give explicitly the value of $\beta$ and the limit. Why should one favour using $\hat{\theta}_{n}$ over $\tilde{\theta}_{n}$ ?