I am trying to prove theorem 7.3.23 in Casella and Burger.
Theorem: Let T be a complete sufficient statistic for a parameter $\theta$, and let $\phi(T)$ be any estimator based only on T. Then $\phi(T)$ is the best unbiased estimator of its expected value.
Here is my attempt to prove this:
To show that $\phi(T)$ is best unbiased estimator for E[$\phi(T)$], I must show that $\phi(T)$ is uncorrelated with any unbiased estimator of $0$. So I must show that $Cov({\phi(T)}, W)=0$ for any W such that W is an unbiased estimator of $0$. For any W such that E[W]=0, $E[E[W|T]]=0$=>$E[W|T]=0$ by completeness of T. So $E[W|T]=0=E[W]$. So W is independent of T. So W and $\phi(T)$ are uncorrelated? So $\phi(T)$ is best unbiased for its expected value. Is this correct?