= Uniformly minimum-variance unbiased estimator
{title2=$\operatorname{Var}_\theta T\leq\operatorname{Var}_\theta U$}
= Uniformly minimum-variance unbiased
{synonym}
= UMVU estimator
{c}
{synonym}
An <unbiased estimator> $T$ is uniformly minimum-variance unbiased if its <variance> is no larger than that of every other <unbiased estimator> of the same parameter, for every parameter value. The <Lehmann–Scheffé theorem> gives such an estimator whenever an unbiased function of a <complete sufficient statistic> exists. Attaining the <Cramér-Rao bound> is sufficient for this property under its hypotheses, but is not necessary: for $T\sim\Gamma(n,\lambda)$ with $n>2$, $(n-1)/T$ is unbiased and has <variance> $\lambda^2/(n-2)$, while the information bound is $\lambda^2/n$. Completeness follows from uniqueness of the <Laplace transform> of $h(t)t^{n-1}$, so the estimator is still uniformly optimal among <unbiased estimators>.
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