Abstract: We investigate statistical inference for functional time series by extending the classic concept of an autocovariance function (ACF) to a functional ACF (FACF). We establish that for functional moving average (FMA) data, the FMA order can be determined as the highest nonvanishing order of an FACF, just as in classic time series analysis. We propose a two-step estimator for the FACF. The first step involves a simultaneous B-spline estimation of each time trajectory, and the second step is a plug-in estimation of the FACF, using the estimated trajectories in place of the latent true curves. Under simple and mild assumptions, the proposed tensor product spline FACF estimator is asymptotically equivalent to the oracle estimator with all known trajectories, leading to an asymptotically correct simultaneous confidence envelope (SCE) for the true FACF. Simulation experiments validate the asymptotic correctness of the SCE and the data-driven FMA order selection. The proposed SCEs are computed for the FACFs of an electroencephalogram (EEG) functional time series, yielding an interesting discovery of a finite FMA lag and Fourier-form functional principal components.

Key words and phrases: Confidence envelope, functional autocovariance function, functional Bartlett's formula, functional moving average, oracle efficiency.