DefinitionDefinition: AI-adaptedProof: Not applicablePipeline-generated

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.

Empirical measure of observed units

Definition

Let $(Z, F)$ be a measurable unit space (Measurable spaces and measurable sets), let $n \geq 1$ , and fix an observed ordered tuple $z_{1}, \dots, z_{n} \in Z$ . Its empirical measure is the set function $P_{n} : = \frac{1}{n} \sum_{i = 1}^{n} δ_{z_{i}}, P_{n} (A) = \frac{1}{n} \sum_{i = 1}^{n} 1 {z_{i} \in A} (A \in F),$ where $δ_{z_{i}}$ is the Dirac set function from The Dirac set function at a point. Repeated observations contribute once for each index at which they occur. The statement that $P_{n}$ is a probability measure is established in The empirical measure is a probability measure ↗.

Depends on

Used by

Size-n bootstrap from the empirical law Algorithm
Pointwise bootstrap need not preserve pair dependence Counterexample
Assignment randomization and bootstrap are different laws Example
Resampling complete cluster records Example
The bootstrap law of a two-point sample mean Example
The empirical measure is a probability measure Lemma
Empirical-law bootstrap draws are conditionally IID Theorem

Dependency tree · two levels

3 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), §§1–2, pp. 1–5 (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)