I'm trying to understand the proof of this theorem.
An unbiased estimator $T$, that is a function of a complete statistic $S$, is unique, i.e. there can't be other unbiased estimators that are functions of that statistic.
Take some arbitrary unbiased estimator $T'$, use the Rao-Blackwell theorem on it with $S$, i.e. the expected value of it conditioning on a sufficient statistic, but take the statistic also complete, $T = \mathbb{E}(T' | S)$
Since it's unique there can only be one of those, and it's variance is better than the estimator you started with, so it's the best (UMVUE).
I have some holes in this proof:
- Does Rao-Blackwell guarantee that $T = \mathbb{E}(T' | S)$ will be ONLY a function of $S$ ? Or do we simply not care if it's also a function of the data, since $S$ is sufficient? Or is the completeness-implying-uniqueness doesn't require it to be ONLY a function of $S$?
- All the proof show is that there won't be another unbiased and complete statistic, but how do we know that there isn't a non-complete unbiased estimator that will be better? I'm missing the point that shows that using the Rao-Blackwell on the complete statistic is not arbitrary. Say I have a complete statistic $S$ and a non-complete statistic $S'$, both being sufficient (as there are infinite sufficient statistics) - will $\mathbb{E}(T' | S)$ always be better than $\mathbb{E}(T' | S')$?