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Milds 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 ${\vec {X}}=X_{1},X_{2},\dots ,X_{n}$ 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 $\varphi :\operatorname {E} [\varphi (Y)]=\theta$ then $\varphi (Y)$ is the unique MVUE of θ.

Proof

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

$\operatorname {E} [\varphi (Y)-\psi (Y)]=0,\theta \in \Omega .$ Since $\{f_{Y}(y:\theta ):\theta \in \Omega \}$ is a complete family

$\operatorname {E} [\varphi (Y)-\psi (Y)]=0\implies \varphi (y)-\psi (y)=0,\theta \in \Omega$ 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 $\operatorname {E} [X]=\theta$ 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 $T=\left(X_{(1)},X_{(n)}\right)$ , the minimal sufficient statistic for $\theta$ (where $X_{(1)}=\min _{i}X_{i}$ and $X_{(n)}=\max _{i}X_{i}$ ), it may be improved using the Rao–Blackwell theorem as follows:

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

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