Getting n=0 after differentiating the log MLE

Question

I'm trying to get the MLE for $\theta$ for the following CDF: $$f_{\theta}=e^{-(x-\theta)}, x>\theta$$

This is what I did: $$L(\theta|X)=\prod_{i=1}^{n}e^{-(x_{i}-\theta)}$$ $$L(\theta|X)=e^{-\sum_{i=1}^{n}(x_{i}-\theta)}$$ $$logL(\theta|X)=-\sum_{i=1}^{n}(x_{i}-\theta)$$ $$logL(\theta|X)=-\sum_{i=1}^{n}(x_{i})+n\theta$$

After $d/d\theta$, I got $n=0$: $$\frac{d}{d\theta}logL(\theta|X)=0$$ $$n=0$$

What is wrong with my computation?

Answer 1

You mean the pdf rather than CDF.

We want to maximize $-\sum_{i=1}^nx_i + n\theta$ subject to $x_i \ge \theta$.

We want $\theta$ to be as large as possible but it has to be smaller than all the $x_i$. Hence $\theta = \min_i x_i$.