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.

Delete-d replicates and subsampling on four units

Example

Take four observed real units $z = (1, 2, 4, 7)$ and use $T_{m} (u_{1}, \dots, u_{m}) = m^{- 1} \sum_{j = 1}^{m} u_{j}$ as the statistic at every nonempty subsample size. For delete-two jackknife replicates, remove each two-unit subset in turn and average the six size-two complement means. For subsampling, draw a two-unit subset uniformly without replacement and compute its mean.

Facts & Assumptions

Given: The ordered tuple $(1, 2, 4, 7)$ and the sample-mean statistic $T_{m} (u_{1}, \dots, u_{m}) = m^{- 1} \sum_{j = 1}^{m} u_{j}$ for $m = 2, 4$ .

[F1]

Delete-d jackknife replicates and their uniform average: the delete- $d$ procedure evaluates the declared size- $(n - d)$ statistic on every complement of a $d$ -subset and returns the uniform average of the replicates; it defines no variance scaling.

[F2]

Uniform subsampling without replacement: the subsampling procedure selects a uniformly distributed subset of the declared size without replacement and applies the corresponding statistic.

[F3]

Partial sums, row sums and sample means defines ${\overline{X}}_{m} = m^{- 1} \sum_{j = 1}^{m} X_{j}$ for real random variables on one probability space. Evaluating this finite-sum identity at an outcome gives the tuple formula declared in Given.

Verification

1.1givenF1F3algebra

The six deleted pairs ${1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 4}, {3, 4}$ leave complement means $11 / 2, 9 / 2, 3, 4, 5 / 2, 3 / 2$ , respectively.

2.1F1F3step 1.1algebra

Their uniform average is $(11 / 2 + 9 / 2 + 3 + 4 + 5 / 2 + 3 / 2) / 6 = 7 / 2$ , equal to the original sample mean; [F1] supplies no multiplier or variance interpretation for that average.

3.1F1F2F3step 1.1step 2.1algebra∎

The six two-unit subsets have means $3 / 2, 5 / 2, 4, 3, 9 / 2, 11 / 2$ , each with probability $1 / 6$ by [F2]. The complement of every two-subset again has size two, and taking the complement twice returns the original subset; hence complementation bijects the deleted pairs with the sampled pairs. Therefore the size-two subsampling law has the same six equally weighted values as the delete-two replicate law in this example. The jackknife output in step 2.1 is their average, a single number, and is not a variance estimate.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

12 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), §10, printed pp. 30–33 (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.J. Bickel, F. Götze and W.R. van Zwet, Resampling Fewer Than n Observations, Statistica Sinica 7 (1997), §3, printed pp. 8–11 (standard reference, not scraped)