Let $X_1, ..., X_n$ be a sample of independent, identically distributed random variables, with density
$$ f_{\theta}(x)=e^{ (\theta -x)}$$.
$x \ge \theta$, otherwise $f_\theta = 0$
The question is: Determine the maximum likelihood estimator $\hat{\theta}_n$ of $\theta$.
I don't understand this question. What exactly does $\hat{\theta}_n$ mean? Wikipedia says something about the nth order statistic:
In statistics, the kth order statistic of a statistical sample is equal to its kth-smallest value. Together with rank statistics, order statistics are among the most fundamental tools in non-parametric statistics and inference.
I tried: $$ L(\theta)=\prod_{i=1}^ne^{ (\theta -x_i)} = e^{(n\theta - \sum_{i=1}^n{x_i})} $$
What's next?