Inference for structural changes in nonstationary functional time series with partial measurement error

Abstract We study the problem of detecting and localizing change points for a general class of locally stationary functional time series. To accommodate the nonstationarity and other possible complex features, such as discontinuous trajectories, and heterogeneous partial measurement error of contemporary functional data, we propose methods that do not rest on the preprocessing techniques of presmoothing and dimension-reduction, which would be less accurate without the assumptions of stationarity and continuous trajectories. For detecting changes, we propose a bootstrap-assisted test for structural breaks among all mean trajectories, which is asymptotically correct and can detect local alternatives of n−1/2. For localizing changes, we develop practical and consistent algorithms for estimating single and multiple change points, which further enable the estimation of mean trajectories. To establish the theoretical properties of our approaches, we develop a new backward martingale difference inequality and a functional Burkholder inequality for nonstationary functional time series, which can be of independent interest. The effectiveness of our approach is demonstrated through extensive simulation studies and real data analyses. The proposed algorithms in this article are available in the R package fcpseed.