Let $X $ be observed data. Let $\hat{\theta}(X)$ be an unbiased estimate of $\theta$ and let T be a sufficient statistic for $\theta$. Define the new estimator $\hat\theta^{*}$ of $\theta$,
$ \hat\theta^{*}(X) =E(\hat\theta(X)| T) $
Then, show that:
- $\hat\theta^{*}(X)$ has a variance that is lower than (or equal to) that of $ \hat\theta$
Hint: for any two random variables $X$ and $Y$, $\operatorname{VAR}(X)= E(\operatorname{VAR}[X|Y]) +VAR[E(X|Y)]$ and $E(E(X|Y)=E(X)$