Definition
Consider estimation of based on data i.i.d. from some member of a family of densities, where is the parameter space. An unbiased estimator of is UMVU if ,
for any other unbiased estimator
If an unbiased estimator of exists, then one can prove there is an essentially unique MVUE. Using the Rao–Blackwell theorem one can also prove that determining the MVUE is simply a matter of finding a complete sufficient statistic for the family 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 UMVU estimator.
Put formally, suppose is unbiased for, and that is a complete sufficient statistic for the family of densities. Then
is the MVUE for
A Bayesian analog is a Bayes estimator, particularly with minimum mean square error (MMSE).
Read more about this topic: Minimum-variance Unbiased Estimator
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