Testing for functional white noise in high dimensions

2026-07-22

AI Summary

This study addresses the lack of effective methods for testing white noise in high-dimensional functional time series by proposing a general testing framework. The approach constructs a supremum-type test statistic based on cross-covariance functions and employs a parametric bootstrap procedure to approximate its null distribution. On the theoretical front, the authors establish a Gaussian approximation result tailored to this setting, rigorously ensuring asymptotic size control and power. The methodology is further extended to accommodate discretely observed functional data and residual diagnostics in functional factor models. Extensive simulations and analyses of two real datasets demonstrate that the proposed test exhibits strong finite-sample performance and practical utility.

Abstract

White noise testing is a fundamental problem in time series analysis. Yet it remains largely unsolved for high-dimensional functional time series, despite the growing attention this area has received in recent years, as existing tests are confined to either univariate functional time series or high-dimensional scalar time series. In this paper, we develop a general error-contamination framework for testing white noise in high-dimensional functional time series. We propose a supremum-type test statistic based on cross-autocovariance functions and develop a parametric bootstrap procedure to approximate its null distribution. By imposing a general high-level condition, we derive a new Gaussian approximation result that ensures size control, and establish an asymptotic power guarantee. We then apply our framework to two concrete applications: (i) white noise test for discretely observed functional time series, and (ii) residual-based goodness-of-fit test for functional factor model. For each problem, we verify the corresponding high-level condition to ensure the theoretical validity of our proposed method. Extensive simulations show that our proposed method achieves good finite-sample performance. The practical utility of our proposed method is further illustrated through applications to two real datasets.

Problem

• white noise testing
• high-dimensional functional time series
• cross-autocovariance
• functional factor model
• goodness-of-fit test

Innovation

• functional white noise
• high-dimensional time series
• cross-autocovariance
• Gaussian approximation
• parametric bootstrap