DefinitionDefinition: 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.

Left empirical quantile of finite draws

Definition

Let $x_{1}, \dots, x_{S} \in R$ with $S \geq 1$ , and let $x_{(1)} \leq \dots \leq x_{(S)}$ be these values in nondecreasing order, retaining multiplicities. For $0 < u \leq 1$ , the left empirical $u$ -quantile is

$Q_{S} (u) : = x_{(⌈ S u ⌉)} .$

Thus ties are retained, and the empirical CDF is $F_{S} (t) = S^{- 1} # {i : x_{i} \leq t}$ . In particular, for $0 < u < 1$ , $Q_{S} (u) = \inf {t : F_{S} (t) \geq u}$ . No interpolation between adjacent order statistics is performed.

Used by

Quantile Monte Carlo standard error Algorithm
Tail effective sample size Algorithm

Dependency tree · 0 levels

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

Sources

Aki Vehtari et al., Rank-normalization, folding, and localization, §4.3 (standard reference, not scraped)