Lehmann–Scheffé theorem

In statistics, the Lehmann–Scheffé theorem is a prominent statement, tying together the ideas of completeness, sufficiency, uniqueness, and best unbiased estimation. The theorem states that any estimator that is unbiased for a given unknown quantity and that depends on the data only through a complete, sufficient statistic is the unique best unbiased estimator of that quantity. The Lehmann–Scheffé theorem is named after Erich Leo Lehmann and Henry Scheffé, given their two early papers.

If ${\displaystyle T}$ is a complete sufficient statistic for ${\displaystyle \theta }$ and ${\displaystyle \operatorname {E} [g(T)]=\tau (\theta )}$ then ${\displaystyle g(T)}$ is the uniformly minimum-variance unbiased estimator (UMVUE) of ${\displaystyle \tau (\theta )}$.