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Conditional K-sample block-bootstrap validity under a joint CLT

Statement

Let $H$ be a real separable Hilbert space. Fix an integer $K \geq 2$ . Let $Y_{i, 1}, \dots, Y_{i, n_{i}} \in H$ , $1 \leq i \leq K$ , be independent functional time-series samples, $M = \sum_{i} n_{i} \to \infty$ , and suppose $n_{i} / M \to θ_{i} \in (0, 1)$ . Under the null hypothesis $μ_{1} = \dots = μ_{K} = μ$ , let ${\overset{ˉ}{Y}}_{i}$ be the sample means and let ${\overset{ˉ}{Y}}_{P} = M^{- 1} \sum_{i} n_{i} {\overset{ˉ}{Y}}_{i}$ . Apply Null-constrained block bootstrap for independent series to these observations with integer block lengths $1 \leq b_{i} \leq n_{i}$ for every sample size used and a specified moving or tapered mode. In tapered mode supply the admissible windows and strictly positive block norms required by Tapered block bootstrap at each $b_{i}$ . Draw the starts conditionally independently across series and blocks, and write ${\overset{ˉ}{Y}}_{i}^{*}$ for the resulting pseudo-series means.

Assume the original centered mean vector

$U_{M} = (\sqrt{n_{i}} ({\overset{ˉ}{Y}}_{i} - μ))_{i = 1}^{K}$

converges weakly to a joint law $L (G_{1}, \dots, G_{K})$ , and assume the conditional law of

$U_{M}^{*} = (\sqrt{n_{i}} ({\overset{ˉ}{Y}}_{i}^{*} - {\overset{ˉ}{Y}}_{P}))_{i = 1}^{K}$

converges conditionally weakly in probability to that same joint law and is conditionally tight in probability: for every $ϵ, η > 0$ there is a deterministic compact $C \subset H^{K}$ such that ${lim sup}_{M} \Pr {\Pr^{*} (U_{M}^{*} \notin C) > ϵ} < η$ . These are explicit joint-limit hypotheses; this theorem does not derive them from the MBB or TBB conditions.

Define the contrast maps

$L_{M} (z_{1}, \dots, z_{K}) = (\sqrt{M / n_{i}} z_{i} - \sqrt{M / n_{K}} z_{K})_{i = 1}^{K - 1}, L_{θ} (z) = (z_{i} / \sqrt{θ_{i}} - z_{K} / \sqrt{θ_{K}})_{i = 1}^{K - 1} .$

For every continuous $Ψ : H^{K - 1} \to S$ into a metric space $S$ , the conditional law of $Ψ (L_{M} U_{M}^{*})$ converges testwise in probability to the law of $Ψ (L_{θ} (G_{1}, \dots, G_{K}))$ : for every bounded continuous $f : S \to R$ , the corresponding conditional expectation tends in probability to $E f (Ψ (L_{θ} (G_{1}, \dots, G_{K})))$ . If $S$ is separable and complete and a weak-convergence metric with measurable conditional distances is supplied, the conclusion is also conditional weak convergence in probability in the sense of Conditional weak convergence in probability. Under the null, the sampling law of $Ψ (L_{M} U_{M})$ has the same limit. The maps may be squared norms, integrated linear contrasts, or any other continuous statistic of this scaled contrast.

Facts & Assumptions

Given: The samples, null, sample-size proportions, joint sampling limit and joint conditional bootstrap limit in the Statement.

[F1]

Null-constrained block bootstrap for independent series: Every pseudo-series has conditional mean equal to the pooled sample mean ${\overset{ˉ}{Y}}_{P}$ .

[F2]

Exact null centering of the K-sample block bootstrap: For each $i$ , $E^{*} {\overset{ˉ}{Y}}_{i}^{*} = {\overset{ˉ}{Y}}_{P}$ .

[F3]

Conditional weak convergence in probability: The conditional law converges in probability in a metric that generates weak convergence.

[F4]

Continuous mapping theorem: Weak convergence is preserved by a measurable map whose discontinuity set has limiting probability zero; in particular this applies to continuous maps.

[F5]

Weak convergence of borel probability measures gives convergence of bounded continuous test integrals. Continuity from above when one set has finite measure gives vanishing tails of the finite random norm $∥ G ∥$ ; the radial-cutoff argument in step 2.1 uses no compact-tightness theorem or additional choice axiom.

[F6]

Subsequence characterization of convergence in probability gives the almost-sure further-subsequence criterion for scalar convergence in probability. Portmanteau theorem gives the closed-set criterion for weak convergence.

Proof

Given: The hypotheses and facts in the Statement and Facts & Assumptions.

1.1F1F2givenalgebra

Since every pseudo-series has conditional mean ${\overset{ˉ}{Y}}_{P}$ by [F1]-[F2], that common location cancels in every contrast. Direct substitution gives $L_{M} U_{M}^{*} = (\sqrt{M} ({\overset{ˉ}{Y}}_{i}^{*} - {\overset{ˉ}{Y}}_{K}^{*}))_{i = 1}^{K - 1}$ and $L_{M} U_{M} = (\sqrt{M} ({\overset{ˉ}{Y}}_{i} - {\overset{ˉ}{Y}}_{K}))_{i = 1}^{K - 1}$ . Thus the two statistics use the same contrast map; under the null the common population mean also cancels from $L_{M} U_{M}$ .

2.1F3F5givenstep 1.1algebra

Endow the finite products with their Hilbert product norms. The coefficients in $L_{M}$ converge to those of $L_{θ}$ , hence $a_{M} = ∥ L_{M} - L_{θ} ∥_{o p} \to 0$ . For $R > 0$ , the function $χ_{R} (z) = \min {1, \max {0, ∥ z ∥ - R}}$ is bounded and continuous, vanishes when $∥ z ∥ \leq R$ , and equals one when $∥ z ∥ \geq R + 1$ . Thus [F5] and the given weak convergence imply ${lim sup}_{M} \Pr (∥ U_{M} ∥ \geq R + 1) \leq E χ_{R} (G) \leq \Pr (∥ G ∥ > R) ⟶ 0 (R \to \infty) .$ The last limit follows from continuity from above since $G$ has finite norm. Given $δ > 0$ , eventually $a_{M} (R + 1) \leq δ$ , so $\Pr (∥ (L_{M} - L_{θ}) U_{M} ∥ > δ) \leq \Pr (∥ U_{M} ∥ > R + 1)$ . First take the limit superior in $M$ and then let $R \to \infty$ . Hence $(L_{M} - L_{θ}) U_{M} \to 0$ in probability, without a compact-tightness or AC premise. For the conditional discrepancy, every supplied compact $C \subset H^{K}$ is norm bounded, say by $R_{C}$ . For arbitrary $ϵ, η > 0$ , conditional tightness supplies such a $C$ with ${lim sup}_{M} \Pr {\Pr^{*} (U_{M}^{*} \notin C) > ϵ} < η$ . Eventually $a_{M} R_{C} \leq δ$ , so $\Pr^{*} (∥ (L_{M} - L_{θ}) U_{M}^{*} ∥ > δ) \leq \Pr^{*} (U_{M}^{*} \notin C)$ . Since $η$ is arbitrary, each conditional discrepancy probability tends to zero in probability.

3.1F3F4F6step 2.1

By [F4], $L_{θ} U_{M} \Rightarrow L_{θ} G$ . For a closed $B \subset H^{K - 1}$ , the sampling probability of $L_{M} U_{M} \in B$ is at most the discrepancy probability from step 2.1 plus $\Pr (L_{θ} U_{M} \in B^{δ})$ , where $B^{δ}$ is the closed $δ$ -neighbourhood. Apply [F6] and let $δ ↓ 0$ to obtain $L_{M} U_{M} \Rightarrow L_{θ} G$ . For the conditional version, combine the input weak-metric distance, truncated at one, with the weighted sum $\sum_{j \geq 1} 2^{- j} \Pr^{*} (∥ (L_{M} - L_{θ}) U_{M}^{*} ∥ > 1 / j)$ . This scalar tends to zero in probability by truncation of the geometric tail and step 2.1. Every subsequence therefore has a further subsequence on which it tends to zero almost surely by [F6]. On each such outcome, [F4] gives weak convergence of the fixed-map conditional laws, and the same closed-neighbourhood inequality proves weak convergence of the varying-map conditional laws to $L (L_{θ} G)$ .

4.1F3F4F6step 3.1∎

Apply [F4] to $Ψ$ for the sampling convergence and, on every almost-sure subsequence outcome of step 3.1, for conditional convergence. Every bounded continuous $f : S \to R$ then has convergent conditional expectations on those outcomes; [F6] gives their convergence in probability along the whole sequence. This proves the stated testwise conclusion for arbitrary metric $S$ . When the supplied output weak-convergence metric is available on separable complete $S$ , the same outcome-wise weak convergence says its distance to the fixed limiting law tends to zero. Applying [F6] to that measurable scalar distance proves the metric conditional conclusion as well.

Remarks

For a statistic expressed as a differentiable functional of the unscaled mean vector, the conditional delta theorem Conditional bootstrap delta method for Hadamard differentiable maps supplies the required Hadamard-differentiability route. The single-series conditional CLTs and tightness bounds are supplied by Moving block bootstrap CLT for dependent Hilbert-valued data and Tapered block bootstrap CLT for dependent Hilbert-valued data under their respective hypotheses. This theorem retains its explicit joint-limit hypotheses; those hypotheses must be checked for the chosen independent-series and resampling construction.

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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), Sections 3.1-3.3, Theorem 3.1 (standard reference, not scraped)