Lehmann-Scheffé theorem

A statisticMathworldMathworldPlanetmath S(𝑿) on a random sample of data 𝑿=(X1,,Xn) is said to be a complete statistic if for any Borel measurable function g,


In other words, g(S)=0 almost everywhere whenever the expected valueMathworldPlanetmath of g(S) is 0. If S(𝑿) is associated with a family f(xθ) of probability density functionsMathworldPlanetmath (or mass function in the discrete case), then completeness of S means that g(S)=0 almost everywhere whenever Eθ(g(S))=0 for every θ.

Theorem 1 (Lehmann-Scheffé).

If S(𝐗) is a complete sufficient statistic and h(𝐗) is an unbiased estimatorMathworldPlanetmath for θ, then, given


h0(S)=h0(S(𝑿)) is a uniformly minimum variance unbiased estimatorMathworldPlanetmath of θ. Furthermore, h0(S) is unique almost everywhere for every θ.

Title Lehmann-Scheffé theorem
Canonical name LehmannScheffeTheorem
Date of creation 2013-03-22 16:31:59
Last modified on 2013-03-22 16:31:59
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 13
Author CWoo (3771)
Entry type Theorem
Classification msc 62F10
Synonym Lehmann-Scheffe theorem
Defines complete statistic