Rao-Blackwell Theorem and UMVUEs

Question

If I use the Rao-Blackwell theorem to find that a conditional statistic has the same variance as the original statistic I conditioned on, does that imply that this statistic is a uniformly minimum variance unbiased estimator for the expected value of that statistic? If not, why not? I'm not quite sure I understand exactly what the Rao-Blackwell theorem implies.

Answer 1

Rao-Blackwell theorem gave us a possibly better unbiased estimator of $\tau(\theta)$ (in the sense of smaller variance) when you have an unbiased estimator of $\tau(\theta)$, named $W$ and a sufficient statistic, named $T$. The estimator $\mathrm{E}(W|T)$ is a better unbiased estimator of $\tau(\theta)$ due to:

(i) Unbiasedness: $\mathrm{E}(\mathrm{E}(W|T))=\tau(\theta)$;

(ii) Efficiency: $\mathrm{Var}(\mathrm{E}(W|T))\leq \mathrm{Var}(W)$;

(iii) A statistic: $\mathrm{E}(W|T)$ is independent of $\theta$ (by sufficiency).

Hence, if an unbiased estimator itself is a function of sufficient statistic, Rao-Blackwell theorem improve none for this estimator. Of course, you cannot say that a sufficient unbiased estimator is always the best unbiased estimator. We need to find another way to improve our unbiased estimator in hand. It is shown that if the sufficient unbiased estimator, e.g. $\mathrm{E}(W|T)$, is correlated with an unbiased estimator of zero, named $U$, $\mathrm{E}(W|T)$ can be always improved by $\mathrm{E}(W|T)+aU$ for some constant $a\in\mathbb{R}$. In contrast, if $\mathrm{E}(W|T)$ is the best unbiased estimator, it must be uncorrelated with $U$, any arbitrary unbiased estimator of $0$. This can be satisfied if $T$ is complete.