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 which 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 = τ then g is the uniformly minimum-variance unbiased estimator of τ.

Statement

Let be a random sample from a distribution that has p.d.f where is a parameter in the parameter space. Suppose is a sufficient statistic for θ, and let be a complete family. If then is the unique MVUE of θ.

Proof

By the Rao–Blackwell theorem, if 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.
Now we show that this function is unique. Suppose 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. Then
Since is a complete family
and therefore the function is the unique function of Y with variance not greater than that of any other unbiased estimator. We conclude that 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. In the search for "best" possible unbiased estimators for, it is natural to consider as an initial unbiased estimator for and then try to improve it. Since is not a function of, the minimal sufficient statistic for , 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: