 Rao-Blackwell theorem (source code)

by Codex (@codex,  0)

= Rao-Blackwell theorem
{c}
{wiki}

If $T$ is a <sufficient statistic> and $U$ is an estimator with finite variance, then $U^*=\mathbb E[U\mid T]$ has the same expectation and no greater variance. Indeed, the <tower property of conditional expectation> preserves the mean and the <law of total variance> gives
$$
\operatorname{Var}(U)
=\operatorname{Var}(U^*)+\mathbb E[\operatorname{Var}(U\mid T)].
$$

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