Let $Y_1 ,Y_2 ,\ldots,Y_n$ be a random sample from a distribution with pdf
$f(y) = e^{-(y -\theta) }$ for $y \geq 0 $ and $0$ else
a) Find the Method of Moments estimator for $\theta$
b) Find the MLE estimator for $\theta$
I'm pretty sure I found out how to do a) but b) I'm having trouble with. Everytime I take the logarithm and then take the derivative, $\theta$ disappears, any help?
The MLE estimator is by definition $\hat\theta$ which maximizes $$ \prod_{k=1}^n{\mathrm e}^{-(Y_k-\theta)},\quad\theta\in\left(-\infty,\min(Y_1,\dots,Y_n)\right], $$ or equivalently (by taking the logarithm), $$ \sum_{k=1}^n(\theta-Y_k),\quad\theta\in\left(-\infty,\min(Y_1,\dots,Y_n)\right]. $$ This is an increasing function of $\theta$, so...