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Functional Sieve Bootstrap for the Partial Sum Process with Application to Change-Point Detection

Efstathios Paparoditis, Lea Wegner, Martin Wendler

TL;DR

This work addresses the problem of inferring the distribution of the partial sum process for weakly stationary functional time series and applying bootstrap methods to change-point detection in the mean. It develops the functional sieve bootstrap (FSB), which projects the process onto a finite Karhunen–Loève representation and fits a finite-order VAR to the score vectors to generate fully functional bootstrap series. The authors prove the bootstrap's validity: the bootstrap partial sum process $Z_{n,m}^*$ converges in probability to the same Brownian motion $W$ with covariance operator $C_W$ as the original process, and the bootstrap-based CUSUM test yields correct asymptotic size, with consistency under suitable local alternatives. Numerical results show that FSB maintains nominal size more reliably than competing resampling methods and demonstrates competitive power across FAR and FMA models with Gaussian and non-Gaussian innovations. Overall, the paper provides a theoretically justified and practically effective tool for functional time-series change-point analysis.

Abstract

This paper applies the functional sieve bootstrap (FSB) to estimate the distribution of the partial sum process for time series stemming from a weakly stationary functional process. Consistency of the FSB procedure under weak assumptions on the underlying functional process is established. This result allows for the application of the FSB procedure to testing for a change-point in the mean of a functional time series using the CUSUM-statistic. We show that the FSB asymptotically correctly estimates critical values of the CUSUM-based test under the null-hypothesis. Consistency of the FSB-based test under local alternatives also is proven. The finite sample performance of the procedure is studied via simulations.

Functional Sieve Bootstrap for the Partial Sum Process with Application to Change-Point Detection

TL;DR

This work addresses the problem of inferring the distribution of the partial sum process for weakly stationary functional time series and applying bootstrap methods to change-point detection in the mean. It develops the functional sieve bootstrap (FSB), which projects the process onto a finite Karhunen–Loève representation and fits a finite-order VAR to the score vectors to generate fully functional bootstrap series. The authors prove the bootstrap's validity: the bootstrap partial sum process converges in probability to the same Brownian motion with covariance operator as the original process, and the bootstrap-based CUSUM test yields correct asymptotic size, with consistency under suitable local alternatives. Numerical results show that FSB maintains nominal size more reliably than competing resampling methods and demonstrates competitive power across FAR and FMA models with Gaussian and non-Gaussian innovations. Overall, the paper provides a theoretically justified and practically effective tool for functional time-series change-point analysis.

Abstract

This paper applies the functional sieve bootstrap (FSB) to estimate the distribution of the partial sum process for time series stemming from a weakly stationary functional process. Consistency of the FSB procedure under weak assumptions on the underlying functional process is established. This result allows for the application of the FSB procedure to testing for a change-point in the mean of a functional time series using the CUSUM-statistic. We show that the FSB asymptotically correctly estimates critical values of the CUSUM-based test under the null-hypothesis. Consistency of the FSB-based test under local alternatives also is proven. The finite sample performance of the procedure is studied via simulations.
Paper Structure (10 sections, 11 theorems, 117 equations, 2 figures, 5 tables)

This paper contains 10 sections, 11 theorems, 117 equations, 2 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Notation and Main Result
  3. The FSB Proposal
  4. Bootstrap validity
  5. Change Point Detection
  6. Behavior under the Null-Hypothesis
  7. Consistency under local alternatives
  8. Numerical Results
  9. Proofs
  10. Further Simulation Results

Key Result

Theorem 2.1

Under Assumptions ass1 and ass2 we have that, as $n\rightarrow\infty$, where $W$ is a Brownian motion in ${\mathcal{H}}$ and the covariance operator of $W(1)$ coincides with the covariance operator $C_W$ given in (eq.CovW).

Figures (2)

  • Figure 1: Size-corrected empirical power for a FAR(1)-process with $C=0.49$ and Gaussian innovations, jump of size $\mu=0.15$ after $100$ of the $n=200$ observations (FSB = functional sieve bootstrap, NBB = non-overlapping block bootstrap, Asymptotic = method by AUE).
  • Figure 2: Size-corrected empirical power for under different models with a jump of $\mu$ after $100$ of $n=200$ observations (FSB = functional sieve bootstrap, NBB = non-overlapping block bootstrap, Asymptotic = method by AUE).

Theorems & Definitions (13)

  • Remark 2.1
  • Theorem 2.1
  • Corollary 3.1
  • Theorem 3.1
  • Corollary 3.2
  • Theorem A.1: Theorem A of SER
  • Theorem A.2: Theorem B of SER
  • Lemma A.1
  • Lemma A.2
  • Lemma A.3
  • ...and 3 more