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

TL;DR

This paper introduces a new weighted testing method for detecting both abrupt and gradual changes in functional time series, addressing limitations of existing dimension reduction and covariance-based approaches.

Contribution

It proposes a novel covariance-weighted test statistic with an offset parameter, applicable to abrupt and gradual change detection in functional data, including asymptotic distributions.

Findings

The new test is scale-invariant and more powerful for certain change types.

Asymptotic distributions under null and alternative hypotheses are derived.

The method performs well in simulations for abrupt and gradual changes.

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.