I am trying to calculate the maximun likelihood estimator, but there is something which is wrong and I can't find the error.
$f(x;\theta)=\theta x^{-2} e^{-\theta/x}$ for x>0.
The likelihood is $\displaystyle\prod_{i=1}^n f(x;\theta) = \displaystyle\prod_{i=1}^n \theta x^{-2} e^{-\theta/x}$ = $\theta^{n} (\displaystyle\sum_{i=1}^n x_{i}^{-2}) e^{-\theta \displaystyle\sum_{i=1}^n x_{i}^{-1}} $
and the log-likelihood is
$l(\theta;x) = log L(\theta;x) = nlog\theta (\displaystyle\sum_{i=1}^n x_{i}^{-2}) - \theta \displaystyle\sum_{i=1}^n x_{i}^{-1}$
To maximize this we differentiate to obtain
$l'(\theta) = \frac{n}{\theta} \displaystyle\sum_{i=1}^n x_{i}^{-2} - \displaystyle\sum_{i=1}^n x_{i}^{-1} $
and then solve $l'(\hat{\theta})=0$ to obtain
$\frac{n}{\theta} \displaystyle\sum_{i=1}^n x_{i}^{-2} = \displaystyle\sum_{i=1}^n x_{i}^{-1} $ $\implies$ $\frac{n}{\hat{\theta}} \frac{1}{\bar{x^2}} = \frac{1}{\bar{x}}$ $\implies$ $\hat{\theta} = \frac{n\bar{x}}{\bar{x^2}} $
I would appreciate if someone could tell me what I am doing wrong.
Thank you.