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.

Finite-sample replacement breakdown

Definition

Let $n \geq 1$ , let $T_{n} : R^{n} \to R$ be a total statistic, and fix a sample $x = (x_{1}, \dots, x_{n})$ . For $0 \leq m \leq n$ , define its maximum replacement displacement by

$B_{m} (T_{n}, x) : = \sup {∣ T_{n} (y) - T_{n} (x) ∣ : y \in R^{n}, # {i : y_{i} \neq x_{i}} \leq m} \in [0, + \infty] .$

When the set is nonempty, the finite-sample replacement breakdown fraction is

$ϵ_{n}^{*} (T_{n}, x) : = \frac{1}{n} \min {m \in {1, \dots, n} : B_{m} (T_{n}, x) = + \infty} .$

Only coordinate replacements are counted, and the replacement values range over all of $R$ . This is a property of the statistic at the fixed sample; it is not a probability or contamination-neighborhood statement.

Used by

Dependency tree · 0 levels

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

Sources

P. B. Stark, Statistics 240 Lecture Notes, Part 9: Robustness and Related Topics, §1.2, PDF pp. 2–3, 11–12 (standard reference, not scraped)
Elvezio Ronchetti, Introduction to Robust Statistics, slides 22–23 (standard reference, not scraped)