I would like to ask a question on a practice problem from a textbook.
The practice problem is about finding estimators of $\theta$, first by using method of moments and then by using a maximum likelihood function, for the following pdf:
$f(x) = \begin{cases} e^{-(x-\theta)}, &\mbox{if }\: x \ge \theta \\ 0, & \mbox{otherwise}. \end{cases} $
To obtain $\hat{\theta}$ via method of moments, I set $\bar{X} = E[X] = \int^{\infty}_\theta x e^{-(\theta -x)}dx = \theta+1,$ giving $\hat{\theta}$ = $\bar{X}-1$.
I am stuck trying to obtain an estimator using a maximum likelihood function. The practice problem says to show that the MLE of $\theta$ is $min(X_i)$.
I set up the following:
$L(\theta ; X_i)=\prod_{i=1}^n e^{-(x_i - \theta)}$
$=e^{-\sum_{i=1}^n (x_i - \theta)}$
Based on a general graph of $y=e^{-x}$, wouldn't one maximise the value of $y$ by minimising the value of $x$? So does that imply we need to minimize $\sum_{i=1}^n (x_i - \theta)$? I am confused here, and would appreciate any tips on how to proceed.
Apologies for any atrocious maths on my part. And thanks in advance.