In my statistical class, we are learning MLE. Last week we end up computing the MLE of parameter $\alpha$ for a random sample $X_1, X_2,...,X_n$ with $\text{Beta}(\alpha,1)$ distribution with support $0<x<1$.
The MLE we found is:
$\hat{\alpha} = - \frac{n}{\log \prod_{i=1}^n X_i }$
Now, I'm trying to compute the mean and variance of this estimator, but have had a hard time trying to figure out how to compute them. The main difficulty comes from having the random variable X_i in the denominator.
Any hint I can consider to get through?
Thanks!