Editing Minimum-variance unbiased estimator (section)

==Definition==
Consider estimation of <math>g(\theta)</math> based on data <math>X_1, X_2, \ldots, X_n</math> i.i.d. from some member of a family of densities <math> p_\theta, \theta \in \Omega</math>, where <math>\Omega</math> is the parameter space. An unbiased estimator <math>\delta(X_1, X_2, \ldots, X_n)</math> of <math> g(\theta) </math> is ''UMVUE'' if <math> \forall \theta \in \Theta</math>,

:<math> \operatorname{var}(\delta(X_1, X_2, \ldots, X_n)) \leq \operatorname{var}(\tilde{\delta}(X_1, X_2, \ldots, X_n)) </math>
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for any other unbiased estimator <math> \tilde{\delta}. </math>

If an unbiased estimator of <math> g(\theta) </math> exists, then one can prove there is an essentially unique MVUE.<ref>{{Cite book|title=U-statistics : theory and practice|last=Lee, A. J., 1946-|date=1990|publisher=M. Dekker|isbn=0824782534|location=New York|oclc=21523971}}</ref> Using the [[Rao–Blackwell theorem]] one can also prove that determining the MVUE is simply a matter of finding a [[complete statistic|complete]] [[sufficient statistic|sufficient]] statistic for the family <math>p_\theta, \theta \in \Omega </math> and conditioning ''any'' unbiased estimator on it.

Further, by the [[Lehmann–Scheffé theorem]], an unbiased estimator that is a function of a complete, sufficient statistic is the UMVUE estimator.

Put formally, suppose <math>\delta(X_1, X_2, \ldots, X_n)</math> is unbiased for <math>g(\theta)</math>, and that <math>T</math> is a complete sufficient statistic for the family of densities. Then

:<math> \eta(X_1, X_2, \ldots, X_n) = \operatorname{E}(\delta(X_1, X_2, \ldots, X_n)\mid T)\,</math>
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is the MVUE for <math>g(\theta). </math>

A [[Bayesian statistics|Bayesian]] analog is a [[Bayes estimator]], particularly with [[minimum mean square error]] (MMSE).