Editing Lehmann–Scheffé theorem (section)

===Proof===

By the [[Rao–Blackwell theorem]], if <math>Z</math> is an unbiased estimator of ''θ'' then <math>\varphi(Y):= \operatorname{E}[Z\mid Y]</math> defines an unbiased estimator of ''θ'' with the property that its variance is not greater than that of <math>Z</math>.

Now we show that this function is unique. Suppose <math>W</math> is another candidate MVUE estimator of ''θ''. Then again <math>\psi(Y):= \operatorname{E}[W\mid Y]</math> defines an unbiased estimator of ''θ'' with the property that its variance is not greater than that of <math>W</math>. Then

:<math>
\operatorname{E}[\varphi(Y) - \psi(Y)] = 0, \theta \in \Omega.
</math>

Since <math>\{ f_Y(y:\theta): \theta \in \Omega\}</math> is a complete family

:<math>
\operatorname{E}[\varphi(Y) - \psi(Y)] = 0 \implies \varphi(y) - \psi(y) = 0, \theta \in \Omega
</math>

and therefore the function <math>\varphi</math> is the unique function of Y with variance not greater than that of any other unbiased estimator. We conclude that <math>\varphi(Y)</math> is the MVUE.