Fully Functional Weighted Testing for Abrupt and Gradual Location Changes in Functional Time Series

TL;DR

This work addresses change-point testing for mean shifts in functional time series within a Hilbert space, introducing a fully functional weighted statistic that incorporates the covariance structure through an offset to achieve scale invariance and enhanced power when changes are not aligned with leading components. The authors derive the null distributions under time dependence for abrupt changes and extend the framework to gradual changes, providing limit theorems and power analyses for three methodological families: dimension-reduction, fully functional, and the proposed weighted approach. They show how the weighted statistic bridges the gap between existing methods, with non-pivotal, covariance-dependent limits requiring estimation of the long-run covariance, and they extend the methodology to gradual change scenarios using weight functions $h$. The results indicate broad applicability and robustness to misspecification, with practical implications for detecting both abrupt and gradual non-stationarities in complex functional data domains.

Abstract

Change point tests for abrupt changes in the mean of functional data, i.e., random elements in infinite-dimensional Hilbert spaces, are either based on dimension reduction techniques, e.g., based on principal components, or directly based on a functional CUSUM (cumulative sum) statistic. The former have often been criticized as not being fully functional and losing too much information. On the other hand, unlike the latter, they take the covariance structure of the data into account by weighting the CUSUM statistics obtained after dimension reduction with the inverse covariance matrix. In this paper, as a middle ground between these two approaches, we propose an alternative statistic that includes the covariance structure with an offset parameter to produce a scale-invariant test procedure and to increase power when the change is not aligned with the first components. We obtain the asymptotic distribution under the null hypothesis for this new test statistic, allowing for time dependence of the data. Furthermore, we introduce versions of all three test statistics for gradual change situations, which have not been previously considered for functional data, and derive their limit distribution. Further results shed light on the asymptotic power behavior for all test statistics under various ground truths for the alternatives.

Theorems & Definitions (16)

• Example 1
• Remark 1
• Remark 2
• Proposition 3.1
• proof
• Proposition A.1
• proof
• proof
• proof : Proof of Theorem \ref{['thm_asymptotics_h0']}
• proof : Proof of Theorem \ref{['thm_alternatives']}