Self-normalization tests for change points in functional time series

Change point detection for functional time series has attracted considerable attention. Existing methods either rely on functional principal component analysis (FPCA), whose finite dimensional projection may perform poorly with complex data, or use bootstrap approaches whose test statistics may be too simple to effectively detect diverse types of changes. We propose a novel self-normalization (SN) test for functional time series implemented via a non-overlapping block bootstrap to circumvent the reliance on FPCA. The test statistic is a normalized cumulative sum (CUSUM) where the normalizing factor allows the capture of subtle local changes in the mean function. The theory contains the weak convergence and test consistency for both the original and the bootstrap versions of the test statistic. We further extend the test to detect changes in the lag-1 autocovariance operator. Simulation studies confirm the superior performance of our test across various settings, and real-world applications further illustrate its practical utility.