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.

Replicate variance for randomized quadrature

Example

Fix the observed data and one randomized quadrature rule for an exact integral $I$ . Suppose its independent full-set randomizations produce IID estimates with mean $I$ and finite variance, and the three realized outputs are $Z_{1} = 2 / 5$ , $Z_{2} = 1 / 2$ , and $Z_{3} = 7 / 10$ . Compute the replicate mean and the variance estimate for that mean. This calculation does not assert a Student interval or estimate variability across new data samples.

Facts & Assumptions

Given: Fixed data and exact integral target $I$ , with IID randomized-rule outputs of mean $I$ and finite variance. The observed outputs are $2 / 5, 1 / 2, 7 / 10$ .

[F1]

For $R \geq 2$ IID replicate outputs, $[R (R - 1)]^{- 1} \sum_{r} (Z_{r} - {\overline{Z}}_{R})^{2}$ is unbiased for the variance of their average. (Unbiased variance estimate from independent computation replicates)

[F2]

A computation stage is recorded as the difference between its exact reference target and its realized output, with all values on the same scalar scale. (Successive statistical and computational error targets)

Verification

1.1givenalgebra

The replicate mean is ${\overline{Z}}_{3} = (2 / 5 + 1 / 2 + 7 / 10) / 3 = 8 / 15$ .

2.1givenstep 1.1algebra

The squared deviations from $8 / 15$ sum to $(2 / 5 - 8 / 15)^{2} + (1 / 2 - 8 / 15)^{2} + (7 / 10 - 8 / 15)^{2} = 7 / 150$ .

3.1F1F2step 1.1step 2.1algebra∎

The replicate variance estimate for the mean is $(3 \cdot 2)^{- 1} (7 / 150) = 7 / 900$ by [F1]. With the data fixed, this estimate concerns only the independent rule-randomization law, not data-sampling variation. The realized computation error in the ledger is ${\overline{Z}}_{3} - I = 8 / 15 - I$ .

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

8 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

Art B. Owen, Monte Carlo Book: The Quasi-Monte Carlo Parts, Chapter 17 §§17.1 and 17.4 (standard reference, not scraped)