I am working through some practice problems before an exam, but I am getting stuck. The PDF is $f(x \mid \theta) = \theta x^{\theta - 1}$, where $0 < x < 1$.
I have already found the MLE as $$\hat{\theta}_{\text{MLE}}=\frac{-n}{\sum_{i = 1}^{n}\log{x_i}}$$
I've also shown that this is both a complete and sufficient statistic, so all I need to do is show that it's unbiased to show that its the UMVUE. This is where I'm drawing a blank. How do I show that it's unbiased?
MLEs are not expected to be unbiased estimators in general. This case is no exception.
So stating the facts to remove this from the unanswered queue:
This $\mathsf{Beta}(\theta,1)$ density is a member of the exponential family, so from Factorisation theorem we directly find that $T = \sum\limits_{i = 1}^n \ln X_i$ is a complete sufficient statistic for $\theta$.
An unbiased estimator of $\theta$ based on $T$ is $h(T) = \dfrac{1 − n}{T}$, which is the UMVUE of $\theta$ by Lehmann-Scheffe theorem. To show this, we can first find the distribution of $T$ and hence calculate $E\left(\dfrac{1}{T}\right)$.
A useful transformation here is the following:
If $X_i\sim \mathsf{Beta}(\theta,1)$, then $-\theta\ln X_i\sim\mathsf{Exp}(1)$ or equivalently, $-2\theta\ln X_i\sim \chi^2_2$.
As the $X_i$'s are i.i.d this means $-2\theta T\sim \chi^2_{2n}$. Or equivalently $T$ has a Gamma distribution (details here).
