Suppose that $\displaystyle Y_{1},...Y_{n}$ constitute a random sample from the density function

$\displaystyle f(y|\theta)=e^{-(y-\theta)}$ for $\displaystyle y>\theta$ and zero elsewhere, where $\displaystyle \theta$ is an unknown, positive constant.

Q: Find an estimator for $\displaystyle \hat{\theta}$ for $\displaystyle \theta$ by the method of maximum likelihood.

A: Consider the likelyhood function $\displaystyle L(\theta)=\Pi_{i=1}^{n}e^{-(y-\theta)}=e^{n\theta\\-\sum_{i=1}^{n}y_{i}}$.

Then,

$\displaystyle ln[L(\theta)]=ln[e^{n\theta\\-\sum_{i=1}^{n}y_{i}}]=n\theta-\sum_{i=1}^{n}y_{i}$.

Thus, the partial derivative with respect to $\displaystyle \theta$ equals $\displaystyle n$.

So, I am doing something wrong, because this attmept is a dead end.