The previous lesson's Remark 2 flagged an uncomfortable fact: the entire raw sample is always trivially sufficient, so "sufficient" alone certifies nothing about how much compression actually happened. This note answers the natural follow-up question — what is the smallest sufficient statistic achievable, the one that every other sufficient statistic can be reduced to? — via a clean likelihood-ratio criterion for minimal sufficiency. It then introduces a second, independent property, completeness, which turns out to be the missing ingredient for one of the sharpest results in classical statistics: Basu's theorem, which delivers exact independence of two statistics essentially for free, without computing a single distribution — reproducing, in a few lines, the same independence of and that Topic 1's Cochran-theorem derivation needed a full moment-generating-function computation to establish.
Learning objectives. By the end of this note you should be able to:
- state the likelihood-ratio criterion for minimal sufficiency and use it to verify that a candidate statistic is minimal sufficient;
- prove that a statistic satisfying the likelihood-ratio criterion is both sufficient and minimal;
- define completeness of a family and verify it directly for the Binomial family by a polynomial argument;
- state Basu's theorem and identify the roles of sufficiency, completeness, and ancillarity in its proof;
- apply Basu's theorem to reprove the independence of and under normality without any moment-generating-function computation.
In this lesson
- Minimal sufficiency
- Completeness
- Basu's theorem
Before you start
- Sufficient Statistics and the Neyman-Fisher Factorization Theorem
- Derivation — Proof of the Factorization Theorem
- Exponential Families
The full lesson, with worked examples and exercise solutions, is for members.
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