LehmannScheffé

The Lehmann–Scheffé theorem is a fundamental result in statistical estimation theory. It identifies when an unbiased estimator can be improved to a uniformly minimum-variance unbiased (UMVU) estimator by conditioning on a complete sufficient statistic.

Formal statement: Let {Pθ} be a statistical model with parameter θ, and suppose T is a complete and sufficient statistic for θ. If W is any unbiased estimator of a parameter function η(θ) (that is, Eθ[W] = η(θ) for all θ), then the conditional expectation Eθ[W | T] is an unbiased estimator of η(θ) that is a function of T. Moreover, among all unbiased estimators that are functions of T, Eθ[W | T] has the minimum variance for every θ. Consequently, Eθ[W | T] is the UMVU estimator of η(θ) when it exists.

Key concepts: Sufficiency means the data can be compressed without loss for estimating θ, as characterized by the factorization theorem. Completeness strengthens sufficiency by ruling out nontrivial estimators with zero expectation for all θ. The theorem is often used with the Rao–Blackwellization idea: improving an unbiased estimator by conditioning on a complete sufficient statistic yields the UMVU estimator.

Implications and uses: The Lehmann–Scheffé theorem provides a practical route to optimal unbiased estimation in many parametric families, particularly exponential families where natural statistics are complete and sufficient.

See also: Rao–Blackwell theorem, unbiased estimation, sufficiency, completeness, UMVU.