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.

[1] 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.

is a complete sufficient statistic for

) ] = τ ( θ )

is the uniformly minimum-variance unbiased estimator (UMVUE) of

τ ( θ )

be a random sample from a distribution that has p.d.f (or p.m.f in the discrete case)

is a parameter in the parameter space.

Suppose

is a sufficient statistic for θ, and let

be a complete family.

is the unique MVUE of θ.

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.

defines an unbiased estimator of θ with the property that its variance is not greater than that of

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.

is the MVUE.

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.

be a random sample from a scale-uniform distribution

and known design parameter

In the search for "best" possible unbiased estimators for

as an initial (crude) unbiased estimator for

, 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: The model is a scale model.

Optimal equivariant estimators can then be derived for loss functions that are invariant.