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Probability, Statistics and Financial Data / Point Estimation: MOM, MLE, and Optimality Theory

UMVUE and the Lehmann-Scheffé Theorem

The Rao-Blackwell theorem left a genuine gap open: conditioning two different unbiased estimators on the same sufficient statistic could, for all Rao-Blackwell's theorem says, produce two different — and possibly incomparable — improved estimators. This note closes that gap completely. Adding the completeness property from Topic 2 to sufficiency turns "an improved estimator" into "the unique best unbiased estimator" — the UMVUE (uniformly minimum-variance unbiased estimator). The Lehmann-Scheffé theorem is the machine that gets you there: given a complete sufficient statistic, any function of it is automatically the UMVUE of its own expectation, and the practical recipe this licenses — find any unbiased estimator, however crude, and condition it on the complete sufficient statistic — turns UMVUE construction from an open-ended search into an algorithm.

This note defines best unbiasedness, proves uniqueness of the best unbiased estimator via Cauchy-Schwarz, states and proves the Lehmann-Scheffé theorem, and works it through the Normal mean, Poisson rate, and Uniform(0,θ)(0,\theta) families — the last one recovering, by a completely different route, exactly the estimator this topic's Cramér-Rao and Rao-Blackwell lessons already singled out.

Learning objectives. By the end of this note you should be able to:
  • define a best unbiased (UMVUE) estimator and prove that, when it exists, it is unique;
  • state and prove the Lehmann-Scheffé theorem, and explain precisely how completeness closes the gap the Rao-Blackwell theorem left open;
  • apply the "find any unbiased estimator, then condition on a complete sufficient statistic" recipe to construct a UMVUE with no obvious candidate in sight;
  • explain why the UMVUE need not attain the Cramér-Rao lower bound, and identify a case where it provably cannot.

In this lesson

  1. Best unbiasedness and its uniqueness
  2. The Lehmann-Scheffé theorem
  3. Worked examples
  4. Method: constructing a UMVUE with no obvious candidate

Before you start

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