Lehmann–Scheffé theorem

In statistics, the Lehmann–Scheffé theorem is 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(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