Minimum Variance Unbiased Estimator

Definition

An estimator $Y=u(X_1,X_2,\dots ,X_n)$ satisfying the following is the minimum variance unbiased estimator (MVUE) for $\theta$ $(E(Y)=\theta )\wedge (\forall T,(E(T)=\theta )\Rightarrow \mathrm{Var}(Y)\le \mathrm{Var}(T))$ where $T$ is an unbiased estimator

An Unbiased Estimator that has lower variance than any other unbiased estimator for the parameter $\theta$

Facts

A minimum variance unbiased estimator does not always exist.

If some unbiased estimator’s variance is equal to the Rao-Cramer Lower Bound for all $\theta$, then it is a minimum variance unbiased estimator.