Lehmann–Scheffé theorem

Lehmann–Scheffé theorem is a fundamental theorem in the field of statistics, specifically in the area of estimation theory. Named after Erich Leo Lehmann and Henry Scheffé, the theorem provides a condition under which a statistical estimator is the unique best unbiased estimator of a parameter.

Etymology

The theorem is named after the statisticians Erich Leo Lehmann and Henry Scheffé, who first proposed it.

Definition

The Lehmann–Scheffé theorem states that if an unbiased estimator of a parameter exists and is complete, then it is the unique best unbiased estimator of that parameter. In other words, no other unbiased estimator has a smaller variance.

Formal Statement

Let T be a sufficient statistic for a parameter θ, and let g(T) be any estimator based on T that is unbiased for a parameter function h(θ). If the statistic T is complete, then g(T) is the unique best unbiased estimator of h(θ).

Proof

The proof of the Lehmann–Scheffé theorem is based on the concept of completeness. By definition, a statistic T is complete if and only if for every function g such that E[g(T)] = 0 for all θ, we have that g(T) = 0 almost surely. This implies that if g(T) and h(T) are both unbiased estimators of the same parameter function, then g(T) - h(T) = 0 almost surely, which means that g(T) = h(T) almost surely.

Applications

The Lehmann–Scheffé theorem is widely used in statistics to find the best unbiased estimator of a parameter. It is particularly useful in the field of parametric statistics, where the form of the underlying distribution is known up to a finite number of parameters.

Related Terms

• Sufficient statistic
• Complete statistic
• Unbiased estimation
• Estimation theory
• Parametric statistics

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