Let $W_1, W_2$ be unbiased estimators of $\theta$ and let $T$ be a sufficient statistic for $\theta$. Is it true that $Var(E(W_1|T)) \leq Var(W_2)$?
I think that it may fails to be true.
If the statement is true then $Var(E(W_1|T))$ is the minimum among the variances $Var(W)$ of all unbiased estimator $W$ so $E(W_1|T)$ is a UMVUE of $\theta$. But Lehmann-Scheffe's theorem tells us that if $T$ is a complete sufficient statistic and $U$ is a unbiased estimator then $E(U|T)$ is a UMVUE of $\theta$. So we don't need the completeness of $T$ if the statement is true.
However I cannot prove it. Anyone can help me?