LemmaStatement: AI-adaptedProof: AI-generatedPipeline-generatedjudge pass (gpt-6-luna)

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AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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Exact null centering of the K-sample block bootstrap

Statement

For fixed data and either mode of Null-constrained block bootstrap for independent series, every pseudo-observation in every series has conditional expectation ${\overset{ˉ}{Y}}_{P}$ . Consequently, for each $i$ ,

$E^{*} {\overset{ˉ}{Y}}_{i}^{*} = {\overset{ˉ}{Y}}_{P},$

so the conditional mean functions of the $K$ pseudo-series are equal.

Facts & Assumptions

Given: The fixed observed series, the block lengths and the independent uniform block-index randomization in the Statement.

[F1]

Null-constrained block bootstrap for independent series: At each series $i$ and within-block position $r$ , the algorithm subtracts $c_{i, r} = N_{i}^{- 1} \sum_{j = 1}^{N_{i}} A_{i, j, r}$ from a uniformly selected $A_{i, J_{i, s}, r}$ , then adds the same pooled mean ${\overset{ˉ}{Y}}_{P}$ .

Proof

Given: The data and pseudo-series construction in the Statement.

1.1F1givenalgebra

Fix $i, s, r$ . Uniformity of $J_{i, s}$ gives $E^{*} A_{i, J_{i, s}, r} - c_{i, r} = \frac{1}{N_{i}} \sum_{j = 1}^{N_{i}} A_{i, j, r} - c_{i, r} = 0$ . Therefore $E^{*} Y_{i, (s - 1) b_{i} + r}^{*} = {\overset{ˉ}{Y}}_{P}$ , including positions in a truncated final block.

2.1step 1.1algebra∎

The bootstrap mean ${\overset{ˉ}{Y}}_{i}^{*}$ is the average of exactly $n_{i}$ such pseudo-observations. Linearity of the finite conditional expectation gives $E^{*} {\overset{ˉ}{Y}}_{i}^{*} = {\overset{ˉ}{Y}}_{P}$ for every $i$ . The common conditional mean proves the stated null imposition.

Depends on

Null-constrained block bootstrap for independent series

Used by

Dependency tree · two levels

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Sources

D. Pilavakis, E. Paparoditis, and T. Sapatinas, Moving block and tapered block bootstrap for functional time series with an application to the K-sample mean problem, Bernoulli 25 (2019), Section 3.2 (standard reference, not scraped)