Minimum-variance unbiased estimator

In statistics a uniformly minimum-variance unbiased estimator or minimum-variance unbiased estimator (UMVUE or MVUE) is an unbiased estimator that has lower variance than any other unbiased estimator for all possible values of the parameter.

For practical statistics problems, it is important to determine the UMVUE if one exists, since less-than-optimal procedures would naturally be avoided, other things being equal. This has led to substantial development of statistical theory related to the problem of optimal estimation. While the particular specification of "optimal" here — requiring unbiasedness and measuring "goodness" using the variance — may not always be what is wanted for any given practical situation, it is one where useful and generally applicable results can be found.

Definition

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