Confidence surfaces for the mean of locally stationary functional time series

The problem of constructing a simultaneous confidence surface for the 2-dimensional mean function of a non-stationary functional time series is challenging as these bands can not be built on classical limit theory for the maximum absolute deviation between an estimate and the time-dependent regression function. In this paper, we propose a new bootstrap methodology to construct such a region. Our approach is based on a Gaussian approximation for the maximum norm of sparse high-dimensional vectors approximating the maximum absolute deviation which is suitable for nonparametric inference of high-dimensional time series. The elimination of the zero entries produces (besides the time dependence) additional dependencies such that the "classical" multiplier bootstrap is not applicable. To solve this issue we develop a novel multiplier bootstrap, where blocks of the coordinates of the vectors are multiplied with random variables, which mimic the specific structure between the vectors appearing in the Gaussian approximation. We prove the validity of our approach by asymptotic theory, demonstrate good finite sample properties by means of a simulation study and illustrate its applicability by analyzing a data example.