AlgorithmAlgorithm: Literature-sourcedProof: 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.

Benjamini–Yekutieli step-up procedure

Definition

Input an integer $m \geq 1$ , observed values $p_{1}, \dots, p_{m} \in [0, 1]$ of p-value statistics (Valid p-values), and a target level $q \in (0, 1)$ . Let $H_{m} = \sum_{j = 1}^{m} 1 / j$ be the harmonic number from Harmonic numbers for set-cover analysis. For $k = 1, \dots, m$ , set

$c_{k} : = \frac{k q}{m H_{m}} .$

Order the pairs $(p_{i}, i)$ lexicographically, so equal values are ordered by their original indices, and write the ordered values $p_{(1)} \leq \dots \leq p_{(m)}$ . Set

$R : = \max ({k \in {1, \dots, m} : p_{(k)} \leq c_{k}} \cup {0}) .$

Reject exactly the hypotheses at ranks $1, \dots, R$ ; when $R = 0$ , reject none. Because $H_{m} \geq 1$ , each critical value satisfies $0 < c_{k} \leq q / H_{m} \leq q < 1$ . This definition specifies the procedure; no FDR guarantee is asserted here.

Depends on

Used by

BY and BH thresholds for independent valid p-values Example
BY controls FDR under arbitrary dependence Theorem

Dependency tree · two levels

6 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

Yoav Benjamini and Daniel Yekutieli, The Control of the False Discovery Rate in Multiple Testing under Dependency, The Annals of Statistics 29(4) (2001), Theorem 1.3 and Section 5 (standard reference, not scraped)
Alexandru Katsevich, STAT 9610 Lecture Notes, Section 6.3.2, Proposition 6.6, printed p. 138 / PDF p. 142 (standard reference, not scraped)