ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedprecheck pass

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.

Resampling complete cluster records

Example

Suppose each of three clusters is recorded as a two-measurement vector, with records $C_{1} = (1, 0)$ , $C_{2} = (1, 1)$ , and $C_{3} = (0, 0)$ . The declared design treats the entire vector as the sampling unit, so a bootstrap draw selects one complete vector at a time.

Facts & Assumptions

Given: Three observed cluster records and a design that declares each vector indivisible for the target analysis.

[F1]

Sampling units and the resampling-unit contract: a resampling unit is a complete unit record or predeclared group selected together; unit compatibility does not assert validity.

[F2]

Size-n bootstrap from the empirical law: each bootstrap draw selects an observed unit record by an independently uniform index.

[F3]

Empirical-law bootstrap draws are conditionally IID: conditional on the fixed unit records, the selected records are IID from their empirical measure.

[F4]

Empirical measure of observed units: each of the three indexed cluster records has empirical mass $1 / 3$ .

Verification

1.1givenF1

The declared records are $C_{1}, C_{2}, C_{3}$ , each a complete ordered pair of measurements, so [F1] makes the vector rather than either coordinate the resampling unit.

1.2F2F3F4algebra

For the index tuple $(2, 2, 1)$ , [F2] returns $(C_{2}, C_{2}, C_{1}) = ((1, 1), (1, 1), (1, 0))$ . By [F2], the three independent index draws each assign probability $1 / 3$ to their specified index, so this index tuple has conditional probability $3^{- 3} = 1 / 27$ . The empirical record law assigns mass $1 / 3$ to each of these three distinct cluster records by [F4], and [F3] identifies the record draws as conditionally IID.

2.1F1F2step 1.2∎

Every bootstrap tuple is formed by selecting records from ${C_{1}, C_{2}, C_{3}}$ , so every selected two-measurement vector remains intact. This verifies the unit contract in this finite example; it makes no claim that resampling clusters is valid for every target or design.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

19 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

Bradley Efron and Robert Tibshirani, The Bootstrap Method for Assessing Statistical Accuracy, Behaviormetrika 12 (1985), §3, printed pp. 7–12 (standard reference, not scraped)
B. Efron and R. Tibshirani, The Bootstrap Method for Assessing Statistical Accuracy, Stanford LCS Technical Report 19 / BIO 101 (October 1985; full-text report copy) (standard reference, not scraped)
P.B. Stark, Statistics 240 Lecture Notes, Part 8: Resampling Methods, §1.1, PDF pp. 2–3 (standard reference, not scraped)