Lehmann–Scheffé theorem

In statistics, the Lehmann–Scheffé theorem ties together completeness, sufficiency, uniqueness, and best unbiased estimation.^[1] 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.^{[2] [3]}

If ${\displaystyle T}${\displaystyle T} is a complete sufficient statistic for ${\displaystyle \theta }${\displaystyle \theta } and ${\displaystyle \operatorname {E} [g(T)]=\tau (\theta )}${\displaystyle \operatorname {E} [g(T)]=\tau (\theta )} then ${\displaystyle g(T)}${\displaystyle g(T)} is the uniformly minimum-variance unbiased estimator (UMVUE) of ${\displaystyle \tau (\theta )}${\displaystyle \tau (\theta )}.

Let ${\displaystyle {\vec {X}}=X_{1},X_{2},\dots ,X_{n}}${\displaystyle {\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) ${\displaystyle f(x:\theta )}${\displaystyle f(x:\theta )} where ${\displaystyle \theta \in \Omega }${\displaystyle \theta \in \Omega } is a parameter in the parameter space. Suppose ${\displaystyle Y=u({\vec {X}})}${\displaystyle Y=u({\vec {X}})} is a sufficient statistic for θ, and let ${\displaystyle \{f_{Y}(y:\theta ):\theta \in \Omega \}}${\displaystyle \{f_{Y}(y:\theta ):\theta \in \Omega \}} be a complete family. If ${\displaystyle \varphi :\operatorname {E} [\varphi (Y)]=\theta }${\displaystyle \varphi :\operatorname {E} [\varphi (Y)]=\theta } then ${\displaystyle \varphi (Y)}${\displaystyle \varphi (Y)} is the unique MVUE of θ.

By the Rao–Blackwell theorem, if ${\displaystyle Z}${\displaystyle Z} is an unbiased estimator of θ then ${\displaystyle \varphi (Y):=\operatorname {E} [Z\mid Y]}${\displaystyle \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 ${\displaystyle Z}${\displaystyle Z}.

Now we show that this function is unique. Suppose ${\displaystyle W}${\displaystyle W} is another candidate MVUE estimator of θ. Then again ${\displaystyle \psi (Y):=\operatorname {E} [W\mid Y]}${\displaystyle \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 ${\displaystyle W}${\displaystyle W}. Then

${\displaystyle \operatorname {E} [\varphi (Y)-\psi (Y)]=0,\theta \in \Omega .}${\displaystyle \operatorname {E} [\varphi (Y)-\psi (Y)]=0,\theta \in \Omega .}

Since ${\displaystyle \{f_{Y}(y:\theta ):\theta \in \Omega \}}${\displaystyle \{f_{Y}(y:\theta ):\theta \in \Omega \}} is a complete family

${\displaystyle \operatorname {E} [\varphi (Y)-\psi (Y)]=0\implies \varphi (y)-\psi (y)=0,\theta \in \Omega }${\displaystyle \operatorname {E} [\varphi (Y)-\psi (Y)]=0\implies \varphi (y)-\psi (y)=0,\theta \in \Omega }

and therefore the function ${\displaystyle \varphi }${\displaystyle \varphi } is the unique function of Y with variance not greater than that of any other unbiased estimator. We conclude that ${\displaystyle \varphi (Y)}${\displaystyle \varphi (Y)} is the MVUE.

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.^[4] Let ${\displaystyle X_{1},\ldots ,X_{n}}${\displaystyle X_{1},\ldots ,X_{n}} be a random sample from a scale-uniform distribution ${\displaystyle X\sim U((1-k)\theta ,(1+k)\theta ),}${\displaystyle X\sim U((1-k)\theta ,(1+k)\theta ),} with unknown mean ${\displaystyle \operatorname {E} [X]=\theta }${\displaystyle \operatorname {E} [X]=\theta } and known design parameter ${\displaystyle k\in (0,1)}${\displaystyle k\in (0,1)}. In the search for "best" possible unbiased estimators for ${\displaystyle \theta }${\displaystyle \theta }, it is natural to consider ${\displaystyle X_{1}}${\displaystyle X_{1}} as an initial (crude) unbiased estimator for ${\displaystyle \theta }${\displaystyle \theta } and then try to improve it. Since ${\displaystyle X_{1}}${\displaystyle X_{1}} is not a function of ${\displaystyle T=\left(X_{(1)},X_{(n)}\right)}${\displaystyle T=\left(X_{(1)},X_{(n)}\right)}, the minimal sufficient statistic for ${\displaystyle \theta }${\displaystyle \theta } (where ${\displaystyle X_{(1)}=\min _{i}X_{i}}${\displaystyle X_{(1)}=\min _{i}X_{i}} and ${\displaystyle X_{(n)}=\max _{i}X_{i}}${\displaystyle X_{(n)}=\max _{i}X_{i}}), it may be improved using the Rao–Blackwell theorem as follows:

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

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

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

The model is a scale model. Optimal equivariant estimators can then be derived for loss functions that are invariant.^[5]

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

• Theorems in statistics
• Estimation theory

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