# Find the Maximum Likelihood Estimator of $\theta$.

Let $X_1,X_2,...,X_n$ be a random sample of size $n$ from a population with density

$f(x) = \left\{ \begin{array}{lr} e^{\theta-x} & , x \geq\theta\\ 0 & , \text{otherwise} \end{array} \right.$

Find the maximum likelihood estimator for $\theta$.

Here is my attempt:

\begin{align*} L(\theta)&=\prod_{k=1}^{n}e^{\theta-x_k}\\ &=\prod_{k=1}^{n}e^\theta e^{-x_k}\\ &=e^{n\theta}\prod_{k=1}^{n}e^{-x_k}\\ &=e^{n\theta} e^{-\sum_{k=1}^{n}} \end{align*}

Then $\ln L(\theta)=n\theta-\sum_{k=1}^{n}x_k$

Take the derivative with respect to $\theta$ to get

$n$

But there is no $\theta$ in this expression, and so how can I find the MLE of $\theta$?