DefinitionDefinition: AI-adaptedProof: Not applicablePipeline-generatedjudge 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.

Orbit upper-tail rank p-values

Definition

For a finite-group randomization setup with statistic $T$ , the inclusive orbit upper-tail rank at $x \in Ω$ is $p_{0} (x) : = \frac{# {g \in G : T (g \cdot x) \geq T (x)}}{∣ G ∣} .$ If $U$ is independent of the observed data and uniform on $[0, 1]$ , the tie-randomized orbit rank is $p_{U} (x, U) : = \frac{# {g \in G : T (g \cdot x) > T (x)} + U # {g \in G : T (g \cdot x) = T (x)}}{∣ G ∣} .$ Both counts are over group elements, with multiplicity. Thus a nonfree action counts a distinct orbit point as many times as it has preimages under $g \mapsto g \cdot x$ .

Depends on

Used by

Without null invariance a group-rank p-value can be anti-conservative Counterexample
Full-orbit and tie-randomized permutation p-values Definition
A permutation orbit with tied ranks and nontrivial stabilizers Example
Exact-orbit and finite-B permutation p-value resolution Example
Exact finite-group permutation p-values and tests Theorem
Finite-group orbit-rank validity Theorem

Dependency tree · two levels

7 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

Peter B. Stark, Statistics 240 Lecture Notes, Part 8: Resampling Methods, §1.3, PDF pp. 10–11 (standard reference, not scraped)
Jesse Hemerik and Jelle Goeman, Exact testing with random permutations, Sections 2–3 (standard reference, not scraped)