Statistical Inference for Functional Time Series

Abstract

Statistical inference is investigated for the mean function of stationary functional time series data with infinite moving average structure. B-spline estimation is proposed for the temporally ordered trajectories of the functional moving average (FMA), 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 which is the sample mean of all trajectories observed entirely without errors. This oracle efficiency allows for the construction of 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 Elec-troEncephalogram (EEG) time series reveals strong evidence of trigonometric form mean function.