I've trying to get the maximum likelihood estimator of $\theta_{MLE}$ but after doing the final derivate step. I've got -1. Am I doing the partial derivative wrongly? What is the MLE of $\theta$?
$$
\begin{equation}
f\left(x|\theta\right)=
\begin{cases}
e^{-(x-\theta)} & x\geq\theta \\
0 & \text{otherwise.}
\end{cases}
\end{equation}
$$
Negative log-likelihood
\begin{split}
L(x|\theta) &= -log(f(x|\theta))\\
&= \sum_{i=1}^n (x_i-\theta)
\end{split}
Partial derivative
\begin{split}
\frac{\partial L(x|\theta)}{\partial\theta} &= \frac{\partial}{\partial\theta}\left(\sum_{i=1}^n (x_i-\theta)\right)
&= -1
\end{split}
Best Answer
The negative log likelihood is $\sum_{i=1}^n (x_i - \theta)$ when $\theta \le \min_i x_i$, and is $\infty$ otherwise. To minimize the negative log likelihood, you should therefore choose $\theta = \min_i x_i$. No need for partial derivatives.