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–Hochberg step-up procedure

Definition

Input $m \geq 1$ p-values $p_{1}, \dots, p_{m} \in [0, 1]$ and a level $q \in (0, 1)$ . Order the hypotheses by the keys $(p_{i}, i)$ , so equal p-values are ordered by their original index, and write the ordered values as $p_{(1)} \leq \dots \leq p_{(m)}$ . Set

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

Reject exactly the hypotheses at ranks $1, \dots, R$ . If the set in the maximum is empty, then $R = 0$ and no hypothesis is rejected. This is a deterministic step-up rule for the supplied p-values; its FDR guarantee is a separate theorem.

Used by

A two-test BH family distinguishes FDR from FWER Example
BY and BH thresholds for independent valid p-values Example
A PRDS bound for each BH true-null contribution Lemma
Leave-one-out identity for the BH step-up rule Lemma
Benjamini–Hochberg FDR under independence Theorem
BH controls FDR under PRDS Theorem

Dependency tree · 0 levels

Nothing. This result depends on no other item in the library.

Sources

Yoav Benjamini and Yosef Hochberg, Controlling the False Discovery Rate: A Practical and Powerful Approach to Multiple Testing, JRSS B 57(1) (1995), §3.1 (standard reference, not scraped)
STATS 200: Introduction to Statistical Inference, Lecture 11: Testing Multiple Hypotheses, Stanford University, PDF pp. 28–29 (standard reference, not scraped)