Abstract: We investigate statistical inference for the mean function of stationary functional time series data with an infinite moving average structure. We propose a B-spline estimation for the temporally ordered trajectories of the functional moving average, which are used to construct a two-step estimator of the mean function. Under mild conditions, the B-spline mean estimator enjoys oracle efficiency in the sense that it is asymptotically equivalent to the infeasible estimator, that is, the sample mean of all trajectories observed entirely without errors. This oracle efficiency allows us to construct a simultaneous confidence band (SCB) for the mean function, which is asymptotically correct. Simulation results strongly corroborate the asymptotic theory. Using the SCB to analyze an electroencephalogram time series reveals strong evidence of a trigonometric form of the mean function.

Key words and phrases: B-spline, electroencephalogram, functional moving average, oracle efficiency, simultaneous confidence band.