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. 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 $\vec= X_1, X_2, \dots, X_n$ 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
*Complete class theorem
*Rao–Blackwell theorem

References

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