Lehmann–Scheffé theorem

In statistics, the Lehmann–Scheffé theorem, named after Erich Leo Lehmann and Henry Scheffé, states that any unbiased estimator based only on a complete, sufficient statistic is the unique best unbiased estimator of its expected value. The Lehmann–Scheffé theorem is prominent in mathematical statistics, tying together the ideas of completeness, sufficiency, uniqueness, and best unbiased estimation.

Formally, if T is a complete sufficient statistic for θ and E(g(T)) = τ(θ) then g(T) is the minimum-variance unbiased estimator (MVUE) of τ(θ).

See also

• Basu's theorem
• Complete class theorem
• Rao–Blackwell theorem

References

• Lehmann, E.L.; Scheffé, H. (1950). "Completeness, similar regions, and unbiased estimation. I.". Sankhyā. 10 (4): 305–340. MR39201. JSTOR 25048038.
• Lehmann, E.L.; Scheffé, H. (1955). "Completeness, similar regions, and unbiased estimation. II". Sankhyā. 15 (3): 219–236. MR72410. JSTOR 25048243.