Change Point Analysis for Functional Data Using Empirical Characteristic Functionals

We develop a new method to detect change points in the distribution of functional data based on integrated CUSUM processes of empirical characteristic functionals. Asymptotic results are presented under conditions allowing for low‐order moments and serial dependence in the data establishing the limiting null‐distribution of the proposed test statistics, as well as their consistency to detect and localize change points in the distribution of functional data. A key consideration in defining these test statistics is the measure used to integrate the CUSUM process over function space. We show that using a measure generated by Brownian motion leads to generally consistent tests. Further, using this measure allows for computationally simple approximations of the necessary integrals, as well as simulation and permutation‐based methods to calibrate detection thresholds for change point analysis. The proposed methods are thoroughly investigated and compared to other existing functional data change point methods in simulation experiments, and are further applied to detect change points in models for continuous electricity demand and high‐frequency asset price returns.