I was also struggled with this statement pointed by the poster: by sufficiency, in the form of the Rao-Blackwell Theorem, we need consider only unbiased estimators of zero based on Y where Y is a complete and sufficient in the context. The authors didn't give much detail explaining why we should only check if the proposed unbiased estimator is uncorrelated with unbiased estimators only based on a complete and sufficient statistic. After some search, I find that an unbiased estimator uncorrelated with unbiased estimator of zero based on sufficient and complete statistic implies the unbiased estimator uncorrelated with all unbiased estimator of zero.
For example, let $h(\tilde{T})$ be an unbiased estimator of $g(\theta)$ based on $\tilde{T}$ which is complete and sufficient. Let $U$ be an unbiased estimator of zero. $U_t$ be unbiased estimator of zero based on $\tilde{T}$. Then the following shows what I claimed above:
Let $\phi_t$=$h(\tilde{T})+cU$ where c is a constant. Here we have $\phi_t$ is an unbiased estimator of zero because:
E($\phi_t$)=E($h(\tilde{T})+cU$)=E($h(\tilde{T})$)+cE($U$)=$g(\theta)$+0=$g(\theta)$
Now we want to know if $h(\tilde{T})$ can be updated by adding random noise (in this case $U$). Since both $h(\tilde{T})$ and $\phi_t$ are unbiased estimators, to decide which one is better, we compare their corresponding variances:
Var($\phi_t$)= Var($h(\tilde{T})+cU$)=Var($h(\tilde{T})$)+$c^2$Var($U$)+2$c$Cov($\tilde{T},U$)
This implies Var($\phi_t$) $\geq$ Var($h(\tilde{T})$) unless Cov($h(\tilde{T}),U$)=0
- Note: Cov($h(\tilde{T}),U$) can be negative so that Var($\phi_t$) $\le$ Var($h(\tilde{T})$)
- Cov($h(\tilde{T}),U$) =E($h(\tilde{T})U$) because E($U$)=0
Here $U$ is all unbiased estimator of zero. The poster asks why $h(\tilde{T})$ uncorrelated with $U_t$ is sufficient in applying theorem 7.3.20. I hope the following answers the questions:
Cov($h(\tilde{T}),U$) =E($h(\tilde{T})U$) = E{E[$h(\tilde{T})U|\tilde{T}$]}=E{$h(\tilde{T})E[U|\tilde{T}]$}=Cov($h(\tilde{T}),E[U|\tilde{T}]$)
where $E[U|\tilde{T}]$ is unbiased estimator of zero based on $\tilde{T}$
So $h(\tilde{T})$ and $E[U|\tilde{T}]$ uncorrelated implies $h(\tilde{T})$ and $U$ uncorrelated.
This is my very first post on StackExchange. Please comment if I made any mistake.