Lehmann–Scheffé theorem

In statistics, the Lehmann–Scheffé theorem ties together 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 $\theta$ and \operatorname (T)\tau(\theta) then $g(T)$ is the uniformly minimum-variance unbiased estimator (UMVUE) of $\tau(\theta)$.

Statement

Let $\vec= X_1, X_2, \dots, X_n$ 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 $Y = u(\vec)$ is a sufficient statistic for ''θ'', and let $\$ be a complete family. If $\varphi:\operatorname$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 \mid Y/math> 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 \mid Y/math> defines an unbiased estimator of ''θ'' with the property that its variance is not greater than that of $W$. Then

:
\operatorname varphi(Y) - \psi(Y)= 0, \theta \in \Omega.

Since $\$ is a complete family

:
\operatorname 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. 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 \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 $T = \left( X_, X_ \right)$, the minimal sufficient statistic for $\theta$ (where $X_ = \min_i X_i$ and $X_ = \max_i X_i$), it may be improved using the Rao–Blackwell theorem as follows:

:\hat_ =\operatorname_\theta _1\mid X_, X_= \frac 2.

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

:$\hat_ = \frac 1 \cdot \frac 2.$

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

:\hat_\text=\frac n \left - \frac \right\frac

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

See also

* Basu's theorem
*Completeness (statistics)
* Rao–Blackwell theorem

References

{{DEFAULTSORT:Lehmann-Scheffe theorem
Theorems in statistics
Estimation theory