DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedprecheck pendingjudge 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.

Complete statistics

Definition

A statistic $T : (X, A) \to (Y, B)$ is complete for $(P_{θ})_{θ \in Θ}$ if, for every measurable real-valued $g$ such that $E_{θ} ∣ g (T) ∣ < \infty$ for every $θ$ , the conditions $E_{θ} [g (T)] = 0$ for every $θ$ imply $P_{θ} (g (T) = 0) = 1$ for every $θ$ . Integrability is part of the premise; completeness is a property of the statistic relative to the specified family.

Depends on

Statistics and estimators

Used by

A curved normal family can make a sufficient statistic incomplete Counterexample
A full exponential family is complete Theorem
Basu’s independence theorem Theorem
Lehmann–Scheffé theorem Theorem

Dependency tree · two levels

4 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)