TheoremStatement: AI-adaptedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-6-luna)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Basu’s independence theorem

Statement

Let $(X, A)$ be standard Borel, and let $T$ and $U$ be statistics from this sample space into standard Borel spaces for a family $(P_{θ})$ . If $T$ is complete and sufficient and $U$ is ancillary, then $T$ and $U$ are independent under every $P_{θ}$ .

Facts & Assumptions

Given: the family on a standard Borel sample space, complete sufficient $T$ , ancillary $U$ , and their standard Borel target spaces.

[F1]

Sufficient statistics: one kernel $K$ gives the conditional law of the data given $T$ for every parameter.

[F2]

Complete statistics: an integrable function of $T$ with zero expectation for every parameter vanishes almost surely under every law.

[F3]

Ancillary statistics: the law of $U$ is parameter-independent.

[F4]

Measure kernel and probability kernel: $y \mapsto K (y, A)$ is measurable and $K (y, \cdot)$ is a probability measure.

Proof

1.1F1F4givenconstruct

Fix a Borel set $C$ in the target of $U$ . The function $q_{C} (y) = K (y, U^{- 1} (C))$ is Borel by [F4]. For every $θ$ , the conditional-law identity [F1] shows $q_{C} (T)$ is a version of $P_{θ} (U \in C ∣ T)$ .

2.1F2F3step 1.1givenalgebra

By ancillarity [F3], $p_{C} = P_{θ} (U \in C)$ is constant in $θ$ . Hence the bounded function $g_{C} (y) = q_{C} (y) - p_{C}$ satisfies $E_{θ} g_{C} (T) = 0$ for every $θ$ . Completeness [F2] implies $q_{C} (T) = p_{C}$ $P_{θ}$ -almost surely for each $θ$ .

3.1F1F3step 2.1givenalgebra∎

For any Borel $D$ in the target of $T$ , $P_{θ} (T \in D, U \in C) = E_{θ} [1_{{T \in D}} q_{C} (T)] = p_{C} P_{θ} (T \in D) = P_{θ} (U \in C) P_{θ} (T \in D)$ . This holds for every pair of Borel sets $D, C$ , which is the definition of independence.

Depends on

Used by

Normal sample mean and residuals are independent Example

Dependency tree · two levels

14 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Ryan Martin, Lecture Notes on Advanced Statistical Theory (standard reference, not scraped)
Joshua M. Tebbs, STAT 512 Mathematical Statistics Lecture Notes (standard reference, not scraped)
Peter Kempthorne, MIT 18.655 Lecture Note 6: Sufficiency (standard reference, not scraped)
Noah Fithian, Stat 210a Reader: Sufficiency (standard reference, not scraped)
Noah Fithian, Stat 210a Reader: Completeness, Ancillarity, and Basu’s Theorem (standard reference, not scraped)
Charles J. Geyer, Stat 5421 Lecture Notes: Exponential Families (standard reference, not scraped)
Jiahao Tian, Chapter 2: Sufficiency and Exponential Families (standard reference, not scraped)