Basu's theorem

Basu's theorem states that an ancillary statistic is independent of a complete sufficient statistic for the same parameter, under the theorem’s regularity conditions.

Sufficient statistic: A statistic that retains all sample information about an unknown parameter under a specified statistical model. Basu's theorem requires one of the two statistics to be sufficient.
Conditional expectation: The expected value of a random variable given specified information or another random variable. The proof studies conditional expectations of functions of the ancillary statistic.
Student's t-distribution: A family of probability distributions used for inference about a normal mean when variance is estimated. In normal samples, the sample mean and sample variance yield a familiar independence result grounded in Basu's theorem.
Rao–Blackwell theorem: A theorem showing that conditioning an estimator on a sufficient statistic cannot increase its variance. It uses sufficiency to improve estimators rather than to prove independence.
Complete statistic: A statistic whose distribution admits no nonzero function with expectation zero for every parameter value. Completeness is the additional condition that makes sufficiency imply independence.
Factorization theorem: A criterion characterizing sufficient statistics through factorization of a model's likelihood. It supplies a standard way to establish the theorem's sufficiency condition.
Normal distribution: A continuous probability distribution with a symmetric bell-shaped density determined by its mean and variance. Normal models provide standard examples of complete sufficient statistics and ancillary statistics.
Cramér–Rao bound: A lower bound on the variance of unbiased estimators under regularity conditions. It constrains estimator variance, unlike Basu's theorem's independence conclusion.
Ancillary statistic: A statistic whose sampling distribution does not depend on the unknown parameter. Ancillarity is the condition imposed on the other statistic in the theorem.
Indicator function: A function equal to one when an event occurs and zero otherwise. Applying completeness to conditional event probabilities yields the independence conclusion.

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Cochran's theoremCompared with: It provides a different route to independence in statistics, based on sufficiency rather than projection matrices.