Gradual Changes in Functional Time Series

We consider the problem of detecting gradual changes in the sequence of mean functions from a not necessarily stationary functional time series. Our approach is based on the maximum deviation (calculated over a given time interval) between a benchmark function and the mean functions at different time points. We speak of a gradual change of size (Formula presented.), if this quantity exceeds a given threshold (Formula presented.). For example, the benchmark function could represent an average of yearly temperature curves from the pre-industrial time, and we are interested in the question of whether the yearly temperature curves afterwards deviate from the pre-industrial average by more than (Formula presented.) degrees Celsius, where the deviations are measured with respect to the sup-norm. Using Gaussian approximations for high-dimensional data, we develop a test for hypotheses of this type and estimators for the time when a deviation of size larger than (Formula presented.) appears for the first time. We prove the validity of our approach and illustrate the new methods by a simulation study and a data example, where we analyze yearly temperature curves at different stations in Australia.