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Lehmann–Scheffé theorem

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Title: Lehmann–Scheffé theorem  
Author: World Heritage Encyclopedia
Language: English
Subject: Erich Leo Lehmann, Rao–Blackwell theorem, Sufficient statistic, Completeness (statistics), Statistical theorems
Collection: Estimation Theory, Statistical Theorems
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Lehmann–Scheffé theorem

In statistics, the Lehmann–Scheffé theorem is prominent statement, tying together the ideas of completeness, sufficiency, uniqueness, and best unbiased estimation.[1] 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.[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).

Contents

  • Statement 1
    • Proof 1.1
  • See also 2
  • References 3

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 \phi:\mathbb{E}[\phi(Y)] = \theta then \phi(Y) is the unique MVUE of θ.

Proof

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

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

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

\mathbb{E}[\phi(Y) - \psi(Y)] = 0 \implies \phi(y) - \psi(y) = 0, \theta \in \Omega

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

See also

References

  1. ^ Casella, George (2001). Statistical Inference. Duxbury Press. p. 369.  
  2. ^  
  3. ^  


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