Lehmann–Scheffé theorem

Short description: Theorem in statistics

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 $\overrightarrow{X}=X_1,X_2,\dots ,X_n$ be a random sample from a distribution that has p.d.f (or p.m.f in the discrete case) $f(x:\theta )$ where $\theta \in \mathrm{\Omega }$ is a parameter in the parameter space. Suppose $Y=u(\overrightarrow{X})$ is a sufficient statistic for θ, and let $\{f_Y(y:\theta ):\theta \in \mathrm{\Omega }\}$ be a complete family. If $\phi :\mathrm{E}[\phi (Y)]=\theta$ then $\phi (Y)$ is the unique MVUE of θ.

Proof

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

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

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

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

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.

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 $X_1,\dots ,X_n$ be a random sample from a scale-uniform distribution $X\sim U((1-k)\theta ,(1+k)\theta ),$ with unknown mean $\mathrm{E}[X]=\theta$ 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 $T=(X_{(1)},X_{(n)})$, the minimal sufficient statistic for $\theta$ (where $X_{(1)}=\underset{i}{\min }{X}_i$ and $X_{(n)}=\underset{i}{\max }{X}_i$), it may be improved using the Rao–Blackwell theorem as follows:

${\hat{\theta }}_{RB}={\mathrm{E}}_{\theta }[X_1\mid X_{(1)},X_{(n)}]=\frac{X_{(1)}+X_{(n)}}{2}.$

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

${\hat{\theta }}_{LV}=\frac{1}{k^2\frac{n-1}{n+1}+1}\cdot \frac{(1-k)X_{(1)}+(1+k)X_{(n)}}{2}.$

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

${\hat{\theta }}_{\text{BAYES}}=\frac{n+1}{n}[1-\frac{\frac{X_{(1)}(1+k)}{X_{(n)}(1-k)}-1}{{\left(\frac{X_{(1)}(1+k)}{X_{(n)}(1-k)}\right)}^{n+1}-1}]\frac{X_{(n)}}{1+k}$

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