I'm having trouble finding the uniform minimum variance unbiased estimator (UMVUE) of non-standard distributions/functions. I understand there are few general approaches that work pretty well, but I'm get still getting stuck. Also, if the UMVUE doesn't exist, how would I show that?
For example. Consider a sample $X_1$, $X_2$, ..., $X_n$ that are iid with density $f(x|\theta) = x \theta exp(-\theta x^2 /2)$
where $x>0$ and $\theta >0$.
Notice, this function is part of the exponential family. Thus, the complete sufficient statistic is a function of $T = \sum_{i=1}^{n} x_i^2$, and the UMVUE is a function of T. The distribution of the sufficient statistic doesn't seem particularly obvious. I want to say it has to do with something like a gamma distribution. There also doesn't seem to be any unbiased estimators that are jumping out, so its hard to know where to go from here. My gut says the UMVUE doesn't exist, but I'm not entirely sure how to show that rigorously.