Lehmann–Scheffé theorem explained

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

\vec{X} = X _{1,} X _{2,} ... , X _{n} be a random sample from a distribution that has p.d.f (or p.m.f in the discrete case) f(x:\theta) where \theta \in \Omega is a parameter in the parameter space. Suppose Y = u(\vec{X}) is a sufficient statistic for θ, and let \{ f _{Y(y:\theta):} \theta \in \Omega\} be a complete family. If \varphi:\operatorname{E}[\varphi(Y)] = \theta then \varphi(Y) is the unique MVUE of θ.

Proof

By the Rao–Blackwell theorem, if

Z is an unbiased estimator of θ then \varphi(Y): = \operatorname{E}[Z\mid Y] 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 \psi(Y): = \operatorname{E}[W\mid Y] defines an unbiased estimator of θ with the property that its variance is not greater than that of W . Then\operatorname{E}[\varphi(Y) - \psi(Y)] = 0 , \theta \in \Omega.

Since

\{ f _{Y(y:\theta):} \theta \in \Omega\} is a complete family\operatorname{E}[\varphi(Y) - \psi(Y)] = 0 \implies \varphi(y) - \psi(y) = 0 , \theta \in \Omega

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.^[1] Let

X _{1,} \ldots, X _{n} be a random sample from a scale-uniform distribution X \sim U ( ( 1- k) \theta, ( 1+ k) \theta), with unknown mean \operatorname{E}[X] = \theta 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 T = \left( X _{(1)} , X _{(n)} \right) , the minimal sufficient statistic for \theta (where X _{(1)} = min _{i} X _{i} and X _{(n)} = max _{i} X _{i} ), it may be improved using the Rao–Blackwell theorem as follows: \hat{\theta} _{RB} = \operatorname{E} _{\theta[X} _{1\mid} X _{(1)} , X _{(} ] = (X _{(1)} + X _{(n)})/(2 .)

However, the following unbiased estimator can be shown to have lower variance:

\hat{\theta} _{LV} = (1)/(k ^{2 (n -1)/(n +1)} +1) ⋅ (( 1- k)X _{(1)} + ( 1+ k) X _{(n)})/(2 .)

And in fact, it could be even further improved when using the following estimator:

\hat{\theta} _{BAYES = (n +1)/(n)} \left[ 1- ((X _{(1)} ( 1+ k))/(-1 X _{(n)} ( 1- k)))/(\left ( (X _{(1)} ( 1+ k))/(X _{(n)} ( 1- k)) \right ) ^{n+1} -1) \right] (X _{(n)})/(1+ k)

The model is a scale model. Optimal equivariant estimators can then be derived for loss functions that are invariant.^[2]

See also

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

Notes and References

1. An Example of an Improvable Rao–Blackwell Improvement, Inefficient Maximum Likelihood Estimator, and Unbiased Generalized Bayes Estimator . Tal Galili . Isaac Meilijson . 31 Mar 2016 . The American Statistician . 70 . 1 . 108–113 . 10.1080/00031305.2015.1100683. 4960505 . 27499547.
2. Taraldsen. Gunnar. 2020. Micha Mandel (2020), "The Scaled Uniform Model Revisited," The American Statistician, 74:1, 98–100: Comment. The American Statistician. 74. 3. 315. 10.1080/00031305.2020.1769727. 219493070 .