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How statement and proof provenance work

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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.

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The bootstrap law of a two-point sample mean

Example

Let $X_{1}, X_{2}$ be IID with $P (X_{i} = 0) = P (X_{i} = 2) = 1 / 2$ , and suppose the observed ordered sample is $(0, 2)$ . The size-two statistic is the sample mean ${\overline{X}}_{2} = (X_{1} + X_{2}) / 2$ from Partial sums, row sums and sample means.

Facts & Assumptions

Given: A possible observed realization $(0, 2)$ from two IID real observations with common law assigning mass $1 / 2$ to each of $0$ and $2$ .

[F1]

Identical distribution and IID families: IID observations have a common law and are mutually independent.

[F2]

Empirical measure of observed units: the empirical measure assigns $P_{n} (A) = \frac{1}{n} \sum_{i} 1 {x_{i} \in A}$ to each measurable set $A$ .

[F3]

Partial sums, row sums and sample means: ${\overline{X}}_{n} = n^{- 1} \sum_{i = 1}^{n} X_{i}$ .

[F4]

Size-n bootstrap from the empirical law: a size- $n$ replicate draws $n$ indices independently and uniformly from the observed indices and returns the corresponding ordered tuple.

[F5]

Empirical-law bootstrap draws are conditionally IID: conditional on fixed observations, each bootstrap entry has law $P_{n}$ and the entries are mutually independent.

Verification

1.1givenF1F2algebra

Under the stated IID law the tuple $(0, 2)$ has probability $(1 / 2) (1 / 2) = 1 / 4$ by [F1], so it is an admissible observed realization. For this fixed tuple, [F2] gives $P_{2} ({0}) = P_{2} ({2}) = 1 / 2$ .

1.2F4F5algebra

By [F4] and [F5], each of the four ordered bootstrap pairs $(0, 0), (0, 2), (2, 0), (2, 2)$ has conditional probability $1 / 4$ .

2.1F3step 1.2algebra∎

Applying the sample-mean formula [F3] to these four pairs gives values $0, 1, 1, 2$ , respectively. Thus the conditional bootstrap mean has probabilities $1 / 4, 1 / 2, 1 / 4$ at $0, 1, 2$ . This is the exact conditional resampling law for this fixed sample and is not a claim of general bootstrap validity.

Depends on

Used by

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Sources

Bradley Efron and Robert Tibshirani, The Bootstrap Method for Assessing Statistical Accuracy, Behaviormetrika 12 (1985), §2, printed pp. 3–5 (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)