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

Statement

Let be a random sample from a distribution that has p.d.f (or p.m.f in the discrete case) f(x:\theta) where is a parameter in the parameter space. Suppose is a sufficient statistic for θ, and let be a complete family. If then \varphi(Y) is the unique MVUE of θ.

Proof

By the Rao–Blackwell theorem, if Z is an unbiased estimator of θ then defines an unbiased estimator of θ with the property that its variance is not greater than that of Z. Now we show that this function is unique. Suppose W is another candidate MVUE estimator of θ. Then again defines an unbiased estimator of θ with the property that its variance is not greater than that of W. Then Since is a complete family and therefore the function \varphi is the unique function of Y with variance not greater than that of any other unbiased estimator. We conclude that \varphi(Y) is the MVUE.

Example for when using a non-complete minimal sufficient statistic

An example of an improvable Rao–Blackwell improvement, when using a minimal sufficient statistic that is not complete, was provided by Galili and Meilijson in 2016. Let be a random sample from a scale-uniform distribution with unknown mean and known design parameter k \in (0,1). In the search for "best" possible unbiased estimators for \theta, it is natural to consider X_1 as an initial (crude) unbiased estimator for \theta and then try to improve it. Since X_1 is not a function of, the minimal sufficient statistic for \theta (where and ), it may be improved using the Rao–Blackwell theorem as follows: However, the following unbiased estimator can be shown to have lower variance: And in fact, it could be even further improved when using the following estimator: The model is a scale model. Optimal equivariant estimators can then be derived for loss functions that are invariant.