$X_1,\ldots , X_n$ i.i.d. and $f_{\theta}(x)=\frac{1}{\theta}x^{\frac{1}{\theta}-1}\hbox{I}_{(0,\infty)}(x)$
Compute $MLE\theta$ and $MSE(\hat \theta)$
I had no problem with computing $MLE\theta$ which is $\hat\theta=-\ln((x_1\cdot\ldots\cdot x_n)^{\frac{1}{n}}) $, but i have a problem now with computing $MSE(\hat\theta)$.
I started with:
$MSE(\hat\theta)=Var(\hat\theta)+(b(\hat\theta))^2$ where $b$ is bias.
But I have a problem even computing a bias.
To be exact, my biggest problem is to compute $\mathbb{E}[\hat\theta]$. I competly don't know what to do, since there is a logarithm in there. Any hints would be appreciated.