ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-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 delete-one jackknife variance for a sample mean

Example

For real data $X_{1}, \dots, X_{n}$ with $n \geq 2$ , take $g (x) = x$ . The delete-one pseudovalues are exactly the observations, and the jackknife variance estimate is the usual sample variance divided by $n$ .

Facts & Assumptions

Given: Real data $X_{1}, \dots, X_{n}$ , $n \geq 2$ , the delete-one definitions, and the estimator $g (x) = x$ , so ${\hat{θ}}_{n} = {\overset{ˉ}{X}}_{n}$ and ${\hat{θ}}_{n, (- i)} = {\overset{ˉ}{X}}_{n, (- i)}$ .

[F1]

For $n \geq 2$ , define ${\overset{ˉ}{θ}}_{n, (-)} = n^{- 1} \sum_{i} {\hat{θ}}_{n, (- i)}$ , $p_{n, i} = n {\hat{θ}}_{n} - (n - 1) {\hat{θ}}_{n, (- i)}$ , and $V_{J, n} = (n - 1) n^{- 1} \sum_{i} ({\hat{θ}}_{n, (- i)} - {\overset{ˉ}{θ}}_{n, (-)})^{2}$ . These are the finite-sample delete-one pseudovalues and jackknife variance; their definitions require no IID or moment assumptions.

[F2]

The delete-one mean is ${\overset{ˉ}{X}}_{n, (- i)} = (n {\overset{ˉ}{X}}_{n} - X_{i}) / (n - 1)$ . IID empirical bootstrap sample and mean statistic

Verification

1.1F2givenalgebra

By [F2], ${\overset{ˉ}{X}}_{n, (- i)} = {\overset{ˉ}{X}}_{n} - (X_{i} - {\overset{ˉ}{X}}_{n}) / (n - 1)$ . Averaging over $i$ gives ${\overset{ˉ}{θ}}_{n, (-)} = {\overset{ˉ}{X}}_{n}$ , since $\sum_{i} (X_{i} - {\overset{ˉ}{X}}_{n}) = 0$ .

2.1F1step 1.1algebra∎

Substitution into [F1] gives $p_{n, i} = X_{i}$ and $V_{J, n} = \frac{n - 1}{n} \sum_{i} {(\frac{{\overset{ˉ}{X}}_{n} - X_{i}}{n - 1})}^{2} = \frac{1}{n (n - 1)} \sum_{i} (X_{i} - {\overset{ˉ}{X}}_{n})^{2} .$ Thus $V_{J, n} = s_{n}^{2} / n$ , where $s_{n}^{2} = (n - 1)^{- 1} \sum_{i} (X_{i} - {\overset{ˉ}{X}}_{n})^{2}$ is the unbiased sample variance.

Depends on

Used by

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Dependency tree · two levels

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Sources

University of Washington STAT 581, Chapter 8: Bootstrap and Jackknife Estimation of Sampling Distributions, §3 (standard reference, not scraped)
Bradley Efron and Robert Tibshirani, The Bootstrap Method for Assessing Statistical Accuracy, Behaviormetrika 12 (1985), §10 (standard reference, not scraped)