Lehmann–Scheffé theorem: Difference between revisions
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{{Short description|Theorem in statistics}} |
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{{Refimprove|date=April 2011}} |
{{Refimprove|date=April 2011}} |
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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.<ref name=Casella/> The theorem states that any [[estimator]] |
In [[statistics]], the '''Lehmann–Scheffé theorem''' is a prominent statement, tying together the ideas of completeness, sufficiency, uniqueness, and best unbiased estimation.<ref name=Casella/> The theorem states that any [[estimator]] that is [[unbiased estimator|unbiased]] for a given unknown quantity and that depends on the data only through a [[completeness (statistics)|complete]], [[sufficiency (statistics)|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.<ref name=LS1/><ref name=LS2/> |
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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''). |
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''). |
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==Statement== |
==Statement== |
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Let <math>\vec{X}= X_1, X_2, \dots, X_n</math> be a random sample from a distribution that has p.d.f (or p.m.f in the discrete case) <math>f(x:\theta)</math> where <math>\theta \in \Omega</math> is a parameter in the parameter space. Suppose <math>Y = u(\vec{X})</math> is a sufficient statistic for ''θ'', and let <math>\{ |
Let <math>\vec{X}= X_1, X_2, \dots, X_n</math> be a random sample from a distribution that has p.d.f (or p.m.f in the discrete case) <math>f(x:\theta)</math> where <math>\theta \in \Omega</math> is a parameter in the parameter space. Suppose <math>Y = u(\vec{X})</math> is a sufficient statistic for ''θ'', and let <math>\{ f_Y(y:\theta): \theta \in \Omega\}</math> be a complete family. If <math>\varphi:\operatorname{E}[\varphi(Y)] = \theta</math> then <math>\varphi(Y)</math> is the unique MVUE of ''θ''. |
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===Proof=== |
===Proof=== |
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By the [[Rao–Blackwell theorem]], if <math>Z</math> is an unbiased estimator of ''θ'' then <math>\ |
By the [[Rao–Blackwell theorem]], if <math>Z</math> is an unbiased estimator of ''θ'' then <math>\varphi(Y):= \operatorname{E}[Z\mid Y]</math> defines an unbiased estimator of ''θ'' with the property that its variance is not greater than that of <math>Z</math>. |
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Now we show that this function is unique. Suppose <math>W</math> is another candidate MVUE estimator of ''θ''. Then again <math>\psi(Y):= \ |
Now we show that this function is unique. Suppose <math>W</math> is another candidate MVUE estimator of ''θ''. Then again <math>\psi(Y):= \operatorname{E}[W\mid Y]</math> defines an unbiased estimator of ''θ'' with the property that its variance is not greater than that of <math>W</math>. Then |
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:<math> |
:<math> |
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\ |
\operatorname{E}[\varphi(Y) - \psi(Y)] = 0, \theta \in \Omega. |
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</math> |
</math> |
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Since <math>\{ |
Since <math>\{ f_Y(y:\theta): \theta \in \Omega\}</math> is a complete family |
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:<math> |
:<math> |
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\ |
\operatorname{E}[\varphi(Y) - \psi(Y)] = 0 \implies \varphi(y) - \psi(y) = 0, \theta \in \Omega |
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</math> |
</math> |
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and therefore the function <math>\ |
and therefore the function <math>\varphi</math> is the unique function of Y with variance not greater than that of any other unbiased estimator. We conclude that <math>\varphi(Y)</math> is the MVUE. |
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== Example for when using a non-complete minimal sufficient statistic == |
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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.<ref>{{cite journal|title= An Example of an Improvable Rao–Blackwell Improvement, Inefficient Maximum Likelihood Estimator, and Unbiased Generalized Bayes Estimator |author1=Tal Galili |author2=Isaac Meilijson | date = 31 Mar 2016 | journal = The American Statistician | volume = 70 | issue = 1 | pages = 108–113 |doi=10.1080/00031305.2015.1100683| pmc = 4960505 | pmid=27499547}}</ref> Let <math>X_1, \ldots, X_n</math> be a random sample from a scale-uniform distribution <math>X \sim U ( (1-k) \theta, (1+k) \theta),</math> with unknown mean <math>\operatorname{E}[X]=\theta</math> and known design parameter <math>k \in (0,1)</math>. In the search for "best" possible unbiased estimators for <math>\theta</math>, it is natural to consider <math>X_1</math> as an initial (crude) unbiased estimator for <math>\theta</math> and then try to improve it. Since <math>X_1</math> is not a function of <math>T = \left( X_{(1)}, X_{(n)} \right)</math>, the minimal sufficient statistic for <math>\theta</math> (where <math>X_{(1)} = \min_i X_i </math> and <math>X_{(n)} = \max_i X_i </math>), it may be improved using the Rao–Blackwell theorem as follows: |
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:<math>\hat{\theta}_{RB} =\operatorname{E}_\theta[X_1\mid X_{(1)}, X_{( n)}] = \frac{X_{(1)}+X_{(n)}} 2.</math> |
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However, the following unbiased estimator can be shown to have lower variance: |
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:<math>\hat{\theta}_{LV} = \frac 1 {k^2\frac{n-1}{n+1}+1} \cdot \frac{(1-k)X_{(1)} + (1+k) X_{(n)}} 2.</math> |
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And in fact, it could be even further improved when using the following estimator: |
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:<math>\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}</math> |
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The model is a [[Scale parameter|scale model]]. Optimal [[Equivariant Estimator|equivariant estimators]] can then be derived for [[loss function]]s that are invariant.<ref>{{Cite journal|last=Taraldsen|first=Gunnar|date=2020|title=Micha Mandel (2020), "The Scaled Uniform Model Revisited," The American Statistician, 74:1, 98–100: Comment|url=https://doi.org/10.1080/00031305.2020.1769727|journal=The American Statistician|volume=74|issue=3|pages=315|doi=10.1080/00031305.2020.1769727|s2cid=219493070 |issn=}}</ref> |
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==See also== |
==See also== |
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*[[Basu's theorem]] |
*[[Basu's theorem]] |
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*[[Completeness (statistics)]] |
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*[[Complete class theorem]] |
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*[[Rao–Blackwell theorem]] |
*[[Rao–Blackwell theorem]] |
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|journal=[[Sankhya (journal)|Sankhyā]] |
|journal=[[Sankhya (journal)|Sankhyā]] |
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|volume=10 |issue=4 |year=1950 |pages=305–340 |
|volume=10 |issue=4 |year=1950 |pages=305–340 |
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|mr=39201 |jstor=25048038}} |
|mr=39201 |jstor=25048038 |doi=10.1007/978-1-4614-1412-4_23|doi-access=free }} |
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</ref> |
</ref> |
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<ref name=LS2>{{cite journal |
<ref name=LS2>{{cite journal |
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|journal=[[Sankhya (journal)|Sankhyā]] |
|journal=[[Sankhya (journal)|Sankhyā]] |
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|volume=15 |issue=3 |year=1955 |pages=219–236 |
|volume=15 |issue=3 |year=1955 |pages=219–236 |
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|mr=72410 |jstor=25048243}} |
|mr=72410 |jstor=25048243 |doi=10.1007/978-1-4614-1412-4_24|doi-access=free }} |
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</ref> |
</ref> |
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<ref name=Casella>{{cite book |
<ref name=Casella>{{cite book |
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|title=Statistical Inference |
|title=Statistical Inference |
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|year=2001 |publisher=Duxbury Press |
|year=2001 |publisher=Duxbury Press |
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|isbn=0-534-24312- |
|isbn=978-0-534-24312-8 |page=369}} |
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</ref> |
</ref> |
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}} |
}} |
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{{Statistics|inference|collapsed}} |
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{{DEFAULTSORT:Lehmann-Scheffe theorem}} |
{{DEFAULTSORT:Lehmann-Scheffe theorem}} |
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[[Category: |
[[Category:Theorems in statistics]] |
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[[Category:Estimation theory]] |
[[Category:Estimation theory]] |
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{{statistics-stub}} |
Latest revision as of 16:14, 9 December 2024
This article needs additional citations for verification. (April 2011) |
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
[edit]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.
Proof
[edit]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 .
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
Since is a complete family
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
[edit]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:
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:
The model is a scale model. Optimal equivariant estimators can then be derived for loss functions that are invariant.[5]
See also
[edit]References
[edit]- ^ 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.