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?

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.
