Lehmann–Scheffé theorem

The Lehmann–Scheffé theorem states that an unbiased estimator measurable with respect to a complete sufficient statistic is the unique uniformly minimum-variance unbiased estimator.

Sufficient statistic: A statistic that preserves all sample information about a parameter under a specified statistical model. Sufficiency ensures the estimator depends on the data through an information-preserving statistic.
Statistical estimator: A rule that maps observed data to an estimate of an unknown parameter or target quantity. The theorem compares estimators of the same target.
Binomial distribution: The distribution of the number of successes in a fixed number of independent trials with a common success probability. The success count is complete and sufficient for the success probability in the fixed-trial model.
Cramér–Rao bound: A lower bound on the variance of unbiased estimators under regularity conditions. It gives a route to proving efficiency, while Lehmann–Scheffé uses completeness and sufficiency.
Erich Leo Lehmann: A German-American statistician known for major contributions to hypothesis testing, estimation, and statistical decision theory. His work with Henry Scheffé established the theorem’s modern form.
Complete statistic: A statistic whose distribution admits no nonzero function with expectation zero for every parameter value. Completeness rules out distinct unbiased functions of the statistic for the same target.
Statistical parameter: A fixed but unknown numerical feature that indexes a statistical model or describes a population. Unbiasedness and variance are evaluated across the model’s parameter values.
Bernoulli distribution: A distribution on zero and one representing a single trial with a specified success probability. A sample of Bernoulli trials yields the binomial statistic used to estimate success probability.
Fisher information: A measure of how much a probability model’s data distribution informs about its parameter. It determines the Cramér–Rao bound, an alternative tool for assessing estimator variance.
Henry Scheffé: An American statistician known for contributions to estimation, analysis of variance, and statistical theory. His joint work with Lehmann gave the result its name.

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Basu's theoremRelated: It uses the same completeness and sufficiency conditions for a different statistical conclusion.