Rao-Blackwell and unbiased estimators of zero In Casella & Berger we are trying to prove that any estimator $\phi$ based on a complete sufficient statistic T is the unique best unbiased estimator of its expectation.
However, in the preceding example, where the "proof" is developed:

(Copied from a website that copied from Casella-Berger; in C&B, example 7.3.22)
The preceding theorem in C&B, 7.3.20, says:

$W$ is the best unbiased estimator of some $\tau(\theta)$ if and only
if $W$ is uncorrelated with all unbiased estimators of 0

Note, this theorem does NOT refer to zero-unbiased estimators based on some sufficient statistics ^
I cannot figure, why do we need to consider only unbiased estimators of zero, which are based on the sufficient statistic $Y$ (statement in parentheses), instead of ALL 0-estimators?
And what does Rao-Blackwell have to do with this?
 A: Rao-Blackwell shows that any estimator that is not based on a sufficient statistic cannot be made worse (and is often improved in practice) by conditioning it upon a sufficient statistic, i.e., converting it to one that is based on a sufficient statistic.  If (and only if) the original estimator is unbiased, the Rao-Blackwellized estimator is also unbiased.  It is therefore sufficient :) to consider only those estimators based on sufficient statistics when looking for a UMVUE, because every estimator that is not in that set has a corresponding estimator in the set that has the same or better performance.
A: It appears that the poster is already aware that the best unbiased estimator will have to be a statistic that will be a function of a sufficient statistic.
Let $U(Y)$ be an unbiased estimator of 0 based on $Y$. In the example the authors want to show $Y$ is the best unbiased estimator of $\theta$. They do this by arguing that $E(U(Y))=0$ implies $U(Y)=0$ a.s (since Y is a complete statistic) and hence $Cov(Y,U(Y))=0$. The poster is wondering: why are the authors only considering unbiased estimators of 0 like $U(Y)$ when Theorem 7.3.20 states that an unbiased estimator is best iff it has zero covariance with ALL unbiased estimators of 0?.
First possible explanation:
Let W be some arbitrary unbiased estimator of $\theta$. If $U$ is an unbiased estimator of 0 then $W+aU$ is also an unbiased estimator of $\theta$. Now let $\phi(Y)=E(W+aU|Y)$ where Y is a sufficient statistic (also complete). By the Rao-Blackwell theorem $Var(\phi(Y)) \leq Var(W+aU)$. Note that $\phi(Y)=E(W|Y)+aE(U|Y)$.
Letting $U(Y)=E(U|Y)$, we can conclude that we should only look at unbiased estimators of 0 based on $Y$.
Second possible explanation:
Let $\phi(T)$ be an unbiased estimator of $\theta$ where $T$ is a sufficient statistic. Now let's look at $\phi(T)+aU$. This addition either yields an estimator based entirely on T or not. If it does not, then we should ignore $\phi(T)+aU$ as a possible candidate for best unbiased estimator. If $\phi(T)+aU$ yields an estimator based on $T$ only say $\phi^*(T)$, then clearly $U$ must be an estimator based on $T$ alone ie $U=\frac{\phi^*(T)-\phi(T)}{a}$. Hence we need to only focus on unbiased estimators of 0 based on $T$
