Suppose $X_1,X_2,\ldots,X_n$ is a random sample drawn from a distribution with pdf $$f_{\theta}(x)=\small\frac{\ln\theta}{\theta-1}\theta^x\,\mathbf1_{0<x<1}\quad,\,\theta>1$$
Does there exist uniformly minimum variance unbiased estimator (UMVUE) of $\theta$? If not, can it be derived which functions of $\theta$ admit a UMVUE ?
The pdf $f_{\theta}$ is a member of the one-parameter exponential family, from which it follows that a complete sufficient statistic for $\theta$ based on the sample is $T=\sum\limits_{k=1}^n X_k$. So of course $T$ is the UMVUE of its expectation which can be found easily.
But I could not find any trivial unbiased estimator of $\theta$ for example, which leads me to ask which functions of $\theta$ are unbiasedly estimable in the first place. If $g(\theta)$ is unbiasedly estimated by some $h(X_1,\ldots,X_n)$, then certainly its UMVUE is given by $E(h\mid T)$.
If I can identify the sampling distribution of $T$ my job would be easier, but I am not certain about going down that road without coming up with any particular transformation that results in a common distribution.
I also wish to know if this distribution has a name, and more importantly, if it is related to a standard distribution (the probability integral transform is not much helpful here).
