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A portmanteau test for multivariate non-stationary functional time series with an increasing number of lags

Lujia Bai, Holger Dette, Weichi Wu

TL;DR

The paper addresses testing for white noise in multivariate locally stationary functional time series without dimension reduction. It introduces a fully nonparametric portmanteau-type statistic that aggregates estimates of functional auto-covariances across an increasing number of lags $s_n$, and derives a high-dimensional Gaussian approximation to enable a difference-based multiplier bootstrap for critical values. Theoretical contributions include a Gaussian approximation framework for degenerate $U$-statistics in a nonstationary functional setting and bootstrap validity under nonstandard limiting behavior, complemented by finite-sample evidence. The approach provides a robust tool for assessing serial dependence in complex, high-dimensional functional data with potential time-varying data-generating mechanisms, with practical applicability to spatio-temporal and other nonstationary contexts.

Abstract

Multivariate locally stationary functional time series provide a flexible framework for modeling complex data structures exhibiting both temporal and spatial dependencies while allowing for time-varying data generating mechanism. In this paper, we introduce a specialized portmanteau-type test tailored for assessing white noise assumptions for multivariate locally stationary functional time series without dimension reduction. A simple bootstrap procedure is proposed to implement the test because the limiting distribution can be non-standard or even does not exist. Our approach is based on a new Gaussian approximation result for a maximum of degenerate $U$-statistics of second-order functional time series, which is of independent interest. Through theoretical analysis and simulation studies, we demonstrate the efficacy and adaptability of the proposed method in detecting departures from white noise assumptions in multivariate locally stationary functional time series.

A portmanteau test for multivariate non-stationary functional time series with an increasing number of lags

TL;DR

The paper addresses testing for white noise in multivariate locally stationary functional time series without dimension reduction. It introduces a fully nonparametric portmanteau-type statistic that aggregates estimates of functional auto-covariances across an increasing number of lags , and derives a high-dimensional Gaussian approximation to enable a difference-based multiplier bootstrap for critical values. Theoretical contributions include a Gaussian approximation framework for degenerate -statistics in a nonstationary functional setting and bootstrap validity under nonstandard limiting behavior, complemented by finite-sample evidence. The approach provides a robust tool for assessing serial dependence in complex, high-dimensional functional data with potential time-varying data-generating mechanisms, with practical applicability to spatio-temporal and other nonstationary contexts.

Abstract

Multivariate locally stationary functional time series provide a flexible framework for modeling complex data structures exhibiting both temporal and spatial dependencies while allowing for time-varying data generating mechanism. In this paper, we introduce a specialized portmanteau-type test tailored for assessing white noise assumptions for multivariate locally stationary functional time series without dimension reduction. A simple bootstrap procedure is proposed to implement the test because the limiting distribution can be non-standard or even does not exist. Our approach is based on a new Gaussian approximation result for a maximum of degenerate -statistics of second-order functional time series, which is of independent interest. Through theoretical analysis and simulation studies, we demonstrate the efficacy and adaptability of the proposed method in detecting departures from white noise assumptions in multivariate locally stationary functional time series.
Paper Structure (21 sections, 14 theorems, 313 equations, 1 figure, 3 tables, 1 algorithm)

This paper contains 21 sections, 14 theorems, 313 equations, 1 figure, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Test for white noise in multivariate locally stationary functional time series
  3. Multivariate locally stationary functional time series
  4. Testing procedures
  5. Main results
  6. Finite sample properties
  7. Selection of tuning parameters
  8. Simulation study
  9. Preliminaries
  10. Proof of Proposition \ref{['prop:esti']}
  11. Auxiliary results
  12. Proof of Proposition \ref{['prop:esti']}
  13. Proof of Theorem \ref{['thm:ga']}
  14. Joint cumulants of locally stationary functional time series
  15. Some auxiliary results
  16. ...and 6 more sections

Key Result

Theorem 3.1

If Assumption ass:expo is satisfied, $N = O(n^{\alpha})$, $\alpha \geq 0$, for some $0 \leq \theta < 1/11$, $s_n^{2} = o((n\tau)^{\theta})$, $\tau \to 0$, $f_n \to 0$, $(\log n)^4/(nb) \to 0$, $M_n > c \log n$, $M_n = o(s_n)$, $s_n/\log n \to \infty$. Then, under the null hypothesis there exists a where the statistic $Q_n$ is defined in eq:testQ.

Figures (1)

  • Figure 1: Simulated rejection probabilities of the test \ref{['hol1']} in model 1 (upper left panel), model 2 (upper right panel) and model 3 (lower panel). The nominal level is $\alpha=0.1$, the sample size is $n=400$ and $N=50$.

Theorems & Definitions (32)

  • Definition 2.1
  • Example 2.1: Multivariate time-varying Karhunen-Loève-type expansion
  • Example 2.2: Multivariate locally stationary iterated functional time series
  • Remark 2.1
  • Remark 2.2
  • Theorem 3.1
  • Remark 3.1
  • Remark 3.2
  • Theorem 3.2
  • Theorem 3.3
  • ...and 22 more