ExampleConstruction: AI-adaptedVerification: AI-adaptedPipeline-generatedprecheck passjudge 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.

The mean has unbounded influence and 1/n breakdown

Example

Let $F$ be any probability law on $R$ with finite first absolute moment $μ = \int y d F (y)$ . Its mean functional has $IF (x) = x - μ$ , so its gross-error sensitivity is infinite. At every fixed sample, its sample mean has replacement breakdown fraction $1 / n$ .

Facts & Assumptions

Given: A probability law $F$ with $\int ∣ y ∣ d F (y) < \infty$ , its mean $μ$ , and a fixed sample of size $n \geq 1$ .

[F1]

Influence function for a regular M-location functional: for $ψ (u) = u$ , the mean's point-contamination influence function is $x - μ$ for every $x \in R$ .

[F2]

Gross-error sensitivity of a functional: $γ^{*} (T, F)$ is the supremum of the absolute influence values on their domain.

[F3]

Replacement breakdown of the sample mean and odd-sample median: the sample mean's replacement breakdown fraction at any fixed sample is $1 / n$ .

Verification

1.1F1

By [F1], the mean influence function exists for every $x \in R$ and equals $x - μ$ .

1.2F3

The sample-mean replacement breakdown fraction is $1 / n$ by [F3], independently of the distribution $F$ .

2.1F2step 1.1algebra

The values $∣ x - μ ∣$ are unbounded as $x$ ranges over $R$ ; hence [F2] gives $γ^{*} (T, F) = + \infty$ .

3.1step 2.1step 1.2∎

Thus in this example the population gross-error sensitivity is infinite, while one arbitrary coordinate replacement already makes the finite-sample mean unbounded.

Depends on

Used by

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Dependency tree · two levels

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Sources

P. B. Stark, Statistics 240 Lecture Notes, Part 9: Robustness and Related Topics, §1.3.4, PDF pp. 16–17 (standard reference, not scraped)
Elvezio Ronchetti, Introduction to Robust Statistics, slides 9–10 and 22 (standard reference, not scraped)