Lehmann–Scheffé theorem

In statistics, the Lehmann–Scheffé theorem is 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 $\vec{X}= X_1, X_2, \dots, X_n$ 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 $\phi:\mathbb{E}[\phi(Y)] = \theta$ then $\phi(Y)$ is the unique MVUE of θ.

Proof

By the Rao–Blackwell theorem, if $Z$ is an unbiased estimator of θ then $\phi(Y):= \mathbb{E}[Z|Y]$ 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 $\psi(Y):= \mathbb{E}[W|Y]$ defines an unbiased estimator of θ with the property that its variance is not greater than that of $W$. Then

$\mathbb{E}[\phi(Y) - \psi(Y)] = 0, \theta \in \Omega.$

Since $\{ f_{Y}(y:\theta): \theta \in \Omega\}$ is a complete family

$\mathbb{E}[\phi(Y) - \psi(Y)] = 0 \implies \phi(y) - \psi(y) = 0, \theta \in \Omega$

and therefore the function $\phi$ is the unique function of Y with variance not greater than that of any other unbiased estimator. We conclude that $\phi(Y)$ is the MVUE.

See also

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

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.