AlgorithmAlgorithm: Literature-sourcedProof: 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.

Null-constrained block bootstrap for independent series

Definition

Input: a fixed integer $K \geq 2$ of observed series $Y_{i, 1}, \dots, Y_{i, n_{i}}$ in a real Hilbert space $H$ , positive integer sample sizes $n_{i}$ , total size $M = \sum_{i = 1}^{K} n_{i}$ , integer block lengths $1 \leq b_{i} \leq n_{i}$ , and a mode, moving or tapered. For tapered mode also supply an admissible window with positive block norm at each $b_{i}$ . Put

${\overset{ˉ}{Y}}_{i} = \frac{1}{n_{i}} \sum_{t = 1}^{n_{i}} Y_{i, t}, {\overset{ˉ}{Y}}_{P} = \frac{1}{M} \sum_{i = 1}^{K} n_{i} {\overset{ˉ}{Y}}_{i}, R_{i, t} = Y_{i, t} - {\overset{ˉ}{Y}}_{i} .$

For each series $i = 1, \dots, K$ , let $N_{i} = n_{i} - b_{i} + 1$ and $k_{i} = ⌈ n_{i} / b_{i} ⌉$ . For every candidate start $j = 1, \dots, N_{i}$ and position $r = 1, \dots, b_{i}$ , define the following arrays. In moving mode set $A_{i, j, r} = R_{i, j + r - 1}$ . In tapered mode set $A_{i, j, r} = a_{b_{i}} w_{b_{i}} (r) R_{i, j + r - 1}$ , with $a_{b_{i}}$ and $w_{b_{i}}$ as in Tapered block bootstrap. For each $i = 1, \dots, K$ and $s = 1, \dots, k_{i}$ , draw independent uniform starts $J_{i, s} \in {1, \dots, N_{i}}$ , using the index law of Overlapping block-index resampling. For each within-block position define

$c_{i, r} = \frac{1}{N_{i}} \sum_{j = 1}^{N_{i}} A_{i, j, r} .$

Output: for each $i$ , a pseudo-series $Y_{i, 1}^{*}, \dots, Y_{i, n_{i}}^{*}$ . In the order $s = 1, \dots, k_{i}$ , retain the positions $r = 1, \dots, q_{i, s}$ , where $q_{i, s} = \min {b_{i}, n_{i} - (s - 1) b_{i}}$ , and set

$Y_{i, (s - 1) b_{i} + r}^{*} = {\overset{ˉ}{Y}}_{P} + A_{i, J_{i, s}, r} - c_{i, r} .$

Thus a divisible sample uses $k_{i}$ complete blocks; otherwise only the first $n_{i} - b_{i} ⌊ n_{i} / b_{i} ⌋$ positions of the final block are retained. This centers each resampled residual position by its own conditional average, then gives every pseudo-series the same pooled location ${\overset{ˉ}{Y}}_{P}$ . All block starts are conditionally independent across blocks and across series.

Remarks

The algorithm is a finite-sample null imposition. Its validity for a statistic requires additional process and block-length hypotheses.

Depends on

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)