Variances of unbiased estimator $W_1$ and $E(W_2|T)$ for another unbiased estimator $W_2$ and a sufficient statistic $T$ 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?  
 A: You can take a look at the proof of Rao-Blackwell theorem,
\begin{align}
var(E[W_1|T])&=E(E[W_1|T] - \theta)^2\\
&=E(E[W_1 - \theta|T])^2\\
&\le E(E(W_1 - \theta)^2|T])\\
&= E(W_1 - \theta)^2=var(W_1).
\end{align}
When the inequality can stem from Jensen's inequality or even from Cauchy-Schwarz inequality. Anyway, it tells you only how to improve a given estimator and without the condition of completeness of $T$, it is not necessarily gives you an UMVUE. 
Thus, your statement is generally false, however it is not that easy to construct a counterexample as in regular cases a sufficient statistic is also complete and the first iteration of the Rao-Blackwell already gives you the UMVUE.       
A: No, the conjecture is not true. If $T$ is sufficient, the unbiased estimator $\mathrm{E}(W_1|T)$ is a function of $T$ and is hence an unbiased estimator of $\theta$ (trivially a statistic by definition of sufficiency). However, if $T$ is not complete, which means it might be correlated with some unbiased estimator of zero, named $U$. Let $W_2= \mathrm{E}(W_1|T) + a U$ be another unbiased estimator of $\theta$. We have
\begin{align*}
\mathrm{Var}(W_2) = \mathrm{Var}(\mathrm{E}(W_1|T)) + a^2\mathrm{Var}(U)
+2a\mathrm{Cov}(\mathrm{E}(W_1|T),U).
\end{align*}
Hence, $\mathrm{Var}(W_2)\leq \mathrm{Var}(\mathrm{E}(W_1|T))$ if
$a^2\mathrm{Var}(U)
+2a\mathrm{Cov}(\mathrm{E}(W_1|T),U)\leq 0$. In other words, 
if $T$ is not complete, we can always find a constant $a$: 
\begin{align*}
a\in\Bigg(0,\frac{-2\mathrm{Cov}(\mathrm{E}(W_1|T),U)}{\mathrm{Var}(U)}\Bigg]; &\quad \mbox{if }Cov(\mathrm{E}(W_1|T),U)<0,\\
a\in\Bigg[\frac{-2\mathrm{Cov}(\mathrm{E}(W_1|T),U)}{\mathrm{Var}(U)},0\Bigg); &\quad \mbox{if }Cov(\mathrm{E}(W_1|T),U)>0,
\end{align*} 
such that $\mathrm{Var}(W_2)\leq \mathrm{Var}(\mathrm{E}(W_1|T))$, where $\mathrm{E}(W_1|T)$ and $W_2$ are all unbiased estimator of $\theta$ and $T$ is sufficient but not complete.
