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 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.[2][3]

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) 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.[4] 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 (crude) unbiased estimator for and then try to improve it. Since is not a function of , the minimal sufficient statistic for (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:

References

1. Casella, George (2001). Statistical Inference. Duxbury Press. p. 369. ISBN 0-534-24312-6.
2. Lehmann, E. L.; Scheffé, H. (1950). "Completeness, similar regions, and unbiased estimation. I.". Sankhyā. 10 (4): 305–340. JSTOR 25048038. MR 39201.
3. Lehmann, E.L.; Scheffé, H. (1955). "Completeness, similar regions, and unbiased estimation. II.". Sankhyā. 15 (3): 219–236. JSTOR 25048243. MR 72410.
4. Tal Galili & Isaac Meilijson (31 Mar 2016). "An Example of an Improvable Rao–Blackwell Improvement, Inefficient Maximum Likelihood Estimator, and Unbiased Generalized Bayes Estimator". The American Statistician. 70 (1): 108–113. doi:10.1080/00031305.2015.1100683.
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