I have the following problem:
Let $X_1,..., X_n$ be a sample of independent, identically distributed random variables, with density $$f_{\theta}(x)=\begin{cases} e^{\theta-x}, & \text{if } x\geq \theta\\ 0, & \text{elsewhere}\\ \end{cases}$$
Let $\hat\theta_n$ be the maximum-likelihood estimator (MLE) of $\theta$. I am asked to find the asymptotic distribution of $\sqrt n (\hat\theta_n-\theta)$ and $\hat\theta_n$ efficiency.
So far, I computed the MLE, which is $\hat\theta_n=min\{x_i\}$, for $i=0,...,n.$. I also know that if I find that $\sqrt n (\hat\theta_n-\theta)$ converges in distribution to $N(0,I(\theta))$, we would have both the asymptotic distribution and efficiency of the estimator.
Central Limit Theorem could be a way to do it, but I'm really struggling on it.
Any hints or solutions would be appreciated.