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In statistics, the Lehmann–Scheffé theorem is a prominent statement, tying together the ideas of 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 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 be a random sample from a distribution that has p.d.f (or p.m.f in the discrete case) where is a parameter in the parameter space. Suppose is a sufficient statistic for θ, and let be a complete family. If then is the unique MVUE of θ.
![{\displaystyle \varphi :\operatorname {E} [\varphi (Y)]=\theta }](https://wikimedia.org/api/rest_v1/media/math/render/svg/1b2ed3d910044552c0ac149c73a68d26be0e327e)
Proof
By the Rao–Blackwell theorem, if is an unbiased estimator of θ then defines an unbiased estimator of θ with the property that its variance is not greater than that of .
![{\displaystyle \varphi (Y):=\operatorname {E} [Z\mid Y]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dfe67921861705df22471b718d0a3a2b187e94d1)
Now we show that this function is unique. Suppose is another candidate MVUE estimator of θ. Then again defines an unbiased estimator of θ with the property that its variance is not greater than that of . Then
![{\displaystyle \psi (Y):=\operatorname {E} [W\mid Y]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f417444c68b112a9011e386f7abb8a5921f62aa9)
![{\displaystyle \operatorname {E} [\varphi (Y)-\psi (Y)]=0,\theta \in \Omega .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ac44dd9a0a5abd0de81787957005deca533a99d9)
Since is a complete family
![{\displaystyle \operatorname {E} [\varphi (Y)-\psi (Y)]=0\implies \varphi (y)-\psi (y)=0,\theta \in \Omega }](https://wikimedia.org/api/rest_v1/media/math/render/svg/a411c74ca8120aa1f240597f516c59a848950faa)
and therefore the function is the unique function of Y with variance not greater than that of any other unbiased estimator. We conclude that 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.[4] Let be a random sample from a scale-uniform distribution with unknown mean and known design parameter . In the search for "best" possible unbiased estimators for , it is natural to consider as an initial (crude) unbiased estimator for and then try to improve it. Since is not a function of , the minimal sufficient statistic for (where and ), it may be improved using the Rao–Blackwell theorem as follows:
![{\displaystyle \operatorname {E} [X]=\theta }](https://wikimedia.org/api/rest_v1/media/math/render/svg/5ff93b91b7c8ec604607cc0916464f8dffd71c3d)
![{\displaystyle {\hat {\theta }}_{RB}=\operatorname {E} _{\theta }[X_{1}\mid X_{(1)},X_{(n)}]={\frac {X_{(1)}+X_{(n)}}{2}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9c899dae33c263d6b85dfe4b98172c97a9fdbe05)
However, the following unbiased estimator can be shown to have lower variance:
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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/24f36dcfe8e40ef3b0c422ad859eacf6d4277456)
The model is a scale model. Optimal equivariant estimators can then be derived for loss functions that are invariant.[5]
See also
References
- ^ Casella, George (2001). Statistical Inference. Duxbury Press. p. 369. ISBN 978-0-534-24312-8.
- ^ Lehmann, E. L.; Scheffé, H. (1950). "Completeness, similar regions, and unbiased estimation. I." Sankhyā. 10 (4): 305–340. doi:10.1007/978-1-4614-1412-4_23. JSTOR 25048038. MR 0039201.
- ^ Lehmann, E.L.; Scheffé, H. (1955). "Completeness, similar regions, and unbiased estimation. II". Sankhyā. 15 (3): 219–236. doi:10.1007/978-1-4614-1412-4_24. JSTOR 25048243. MR 0072410.
- ^ Tal Galili; Isaac Meilijson (31 Mar 2016). "An Example of an Improvable Rao–Blackwell Improvement, Inefficient Maximum Likelihood Estimator, and Unbiased Generalized Bayes Estimator". The American Statistician. 70 (1): 108–113. doi:10.1080/00031305.2015.1100683. PMC 4960505. PMID 27499547.
- ^ Taraldsen, Gunnar (2020). "Micha Mandel (2020), "The Scaled Uniform Model Revisited," The American Statistician, 74:1, 98–100: Comment". The American Statistician. 74 (3): 315. doi:10.1080/00031305.2020.1769727. S2CID 219493070.
