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 Δ$$ \Delta $$, if this quantity exceeds a given threshold Δ>0$$ \Delta >0 $$. 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 Δ=1.5$$ \Delta =1.5 $$ 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 Δ$$ \Delta $$ 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.