Lehmann–Scheffé theorem

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

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).

Let ${\vec {X}}=X_{1},X_{2},\dots ,X_{n}$ $\vec{X}= X_1, X_2, \dots, X_n$ be a random sample from a distribution that has p.d.f (or p.m.f in the discrete case) $f(x:\theta )$ $f(x:\theta )$ where $\theta \in \Omega$ $\theta \in \Omega$ is a parameter in the parameter space. Suppose $Y=u({\vec {X}})$ $Y=u({\vec {X}})$ is a sufficient statistic for θ, and let $\{f_{Y}(y:\theta ):\theta \in \Omega \}$ $\{f_{Y}(y:\theta ):\theta \in \Omega \}$ be a complete family. If $\varphi :\operatorname {E} [\varphi (Y)]=\theta$ $\varphi :\operatorname {E} [\varphi (Y)]=\theta$ then $\varphi (Y)$ $\varphi (Y)$ is the unique MVUE of θ.

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