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Instatistics, theLehmann–Scheffé theorem ties together completeness, sufficiency, uniqueness, and best unbiased estimation.^[1] The theorem states that anyestimator that isunbiased for a given unknown quantity and that depends on the data only through acomplete,sufficient statistic is the uniquebest unbiased estimator of that quantity. The Lehmann–Scheffé theorem is named afterErich Leo Lehmann andHenry Scheffé, given their two early papers.^[2][3]

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

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

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

Now we show that this function is unique. Suppose${\displaystyle W}$ is another candidate MVUE estimator ofθ. Then again${\displaystyle \psi (Y):=\mathrm{E}[W\mid Y]}$ defines an unbiased estimator ofθ with the property that its variance is not greater than that of${\displaystyle W}$. Then

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

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

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

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

An example of an improvable Rao–Blackwell improvement, when using a minimal sufficient statistic that isnot complete, was provided by Galili and Meilijson in 2016.^[4] Let${\displaystyle X_1,\dots ,X_n}$ be a random sample from a scale-uniform distribution${\displaystyle X\sim U((1-k)\theta ,(1+k)\theta ),}$ with unknown mean${\displaystyle \mathrm{E}[X]=\theta }$ and known design parameter${\displaystyle k\in (0,1)}$. In the search for "best" possible unbiased estimators for${\displaystyle \theta }$, it is natural to consider${\displaystyle X_1}$ as an initial (crude) unbiased estimator for${\displaystyle \theta }$ and then try to improve it. Since${\displaystyle X_1}$ is not a function of${\displaystyle T=\left(X_{(1)},X_{(n)}\right)}$, the minimal sufficient statistic for${\displaystyle \theta }$ (where${\displaystyle X_{(1)}=\underset{i}{min}{X}_i}$ and${\displaystyle X_{(n)}=\underset{i}{max}{X}_i}$), it may be improved using the Rao–Blackwell theorem as follows:

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

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

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

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

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

The model is ascale model. Optimalequivariant estimators can then be derived forloss functions that are invariant.^[5]

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

• Theorems in statistics
• Estimation theory

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