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Finite-index bootstrap draws are conditionally IID from the empirical law

Statement

For fixed data $x_{1}, \dots, x_{n} \in R^{d}$ , the coordinates of the finite-index bootstrap sample in IID empirical bootstrap sample and mean statistic are independent and each has law $P_{n}$ .

Facts & Assumptions

Given: The fixed data, empirical law, and finite product index space in the definition.

[F1]

The bootstrap index tuple is uniform on ${1, \dots, n}^{n}$ , and $X_{j}^{*} = x_{I_{j}}$ . IID empirical bootstrap sample and mean statistic

[F2]

$P_{n}$ is a probability measure. A finite empirical law is a probability measure

[F3]

Measurable functions of independent random elements remain independent. Measurable coordinatewise functions preserve independence

Proof

1.1F1givenalgebra

For subsets $C_{1}, \dots, C_{n} \subseteq {1, \dots, n}$ , the product-uniform index law gives $P^{*} (I_{1} \in C_{1}, \dots, I_{n} \in C_{n}) = \frac{\prod_{j = 1}^{n} ∣ C_{j} ∣}{n^{n}} = \prod_{j = 1}^{n} \frac{∣ C_{j} ∣}{n} .$ Thus the coordinate indices are independent and each is uniform on ${1, \dots, n}$ .

2.1F1F2F3given∎

For each $j$ , the map $i \mapsto x_{i}$ from the finite discrete index set to $R^{d}$ is measurable, and its pushforward of the uniform law assigns $A \in B (R^{d})$ mass $n^{- 1} \sum_{i} 1 {x_{i} \in A} = P_{n} (A)$ . Applying [F3] to the independent indices shows the values $X_{j}^{*} = x_{I_{j}}$ are independent; their common law is $P_{n}$ , which is a probability law by [F2].

Depends on

Used by

Discrete median bootstrap with ties Example
Conditional IID bootstrap CLT for an empirical law Theorem

Cited to discharge well-definedness by IID empirical bootstrap sample and mean statistic.

Dependency tree · two levels

19 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

University of Washington STAT 581, Chapter 8: Bootstrap and Jackknife Estimation of Sampling Distributions, §2 (standard reference, not scraped)
Bradley Efron and Robert Tibshirani, The Bootstrap Method for Assessing Statistical Accuracy, Behaviormetrika 12 (1985), §2 (standard reference, not scraped)