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Exact conditional covariance of a moving block bootstrap mean

Statement

Fix a positive integer $n$ , observations $x_{1}, \dots, x_{n} \in H$ , and an integer block length $b$ with $1 \leq b \leq n$ . Put $N = n - b + 1$ , $k = ⌊ n / b ⌋$ , and $r = n - k b$ . For $q \in {b}$ , and also $q = r$ when $r > 0$ , define

$S_{j}^{(q)} = \sum_{ℓ = 1}^{q} x_{j + ℓ - 1}, {\overset{ˉ}{S}}_{q} = N^{- 1} \sum_{j = 1}^{N} S_{j}^{(q)}, C_{q} = N^{- 1} \sum_{j = 1}^{N} (S_{j}^{(q)} - {\overset{ˉ}{S}}_{q}) \otimes (S_{j}^{(q)} - {\overset{ˉ}{S}}_{q}) .$

Set $C_{0} = 0$ . If ${\overset{ˉ}{X}}_{n}^{*}$ is the mean produced by Moving block bootstrap, then its conditional covariance operator is

$n {Cov}^{*} ({\overset{ˉ}{X}}_{n}^{*}) = \frac{k C_{b} + C_{r}}{n},$

where the $C_{r}$ term is zero when $r = 0$ . Here ${Cov}^{*} (Z) = E^{*} [(Z - E^{*} Z) \otimes (Z - E^{*} Z)]$ .

Facts & Assumptions

Given: Fixed $H$ -valued observations, block size $b$ , and the independent uniform block-start randomization.

[F1]

Moving block bootstrap: It draws independent uniform block starts, concatenates the corresponding consecutive blocks, and truncates only the last block to produce exactly $n$ observations.

Proof

Given: The observations and the bootstrap law in the Statement.

1.1F1givenalgebra

Write $n = k b + r$ , $0 \leq r < b$ . The sum of the first $n$ bootstrap observations is the sum of $k$ complete selected block sums and, if $r > 0$ , the first $r$ entries of one additional selected block. Conditional on the data, these $k + 1_{{r > 0}}$ contributions are independent, and the centered covariance of each complete contribution is $C_{b}$ , while that of the partial contribution is $C_{r}$ by uniformity of its start.

2.1step 1.1algebra∎

The covariance of a sum of independent centered Hilbert-valued random elements is the sum of their covariance operators: expanding the tensor product leaves only diagonal terms, since each off-diagonal expected tensor is the tensor of a zero mean with a zero mean. Dividing the sum by $n$ to obtain ${\overset{ˉ}{X}}_{n}^{*}$ , and multiplying its covariance by $n$ , gives $(k C_{b} + C_{r}) / n$ .

Remarks

When $r = 0$ , this reduces to $n {Cov}^{*} ({\overset{ˉ}{X}}_{n}^{*}) = C_{b} / b$ . The identity is finite-sample and makes no stationarity or moment assumption on an underlying process.

Depends on

Moving block bootstrap

Used by

Moving block bootstrap CLT for dependent Hilbert-valued data 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

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 2.2 (standard reference, not scraped)