Abstract: Abstract: Statistical inference is developed for vector-valued functional panel data which are i.i.d. with respect to subjects and infinite moving average in time. B-spline estimation is proposed for trajectories, which are used to construct a two-step estimator of the vector mean function. By using explicit Gaussian strong approximation in vector form, in the context of moving average panel, the proposed spline estimator is shown to be oracally efficient in the sense that it is asymptotically equivalent to the infeasible estimator with all trajectories known. This deep theoretical result points to a limiting Gaussian distribution of the vector mean estimator, which allows for the construction of various simultaneous confidence region (SCR) for the vector mean function itself and linear combination of its elements. Asymptotic correctness of the SCRs is both established in theory and validated in simulation experiments. The proposed SCRs are applied to an Electroencephalogram (EEG) multivariate functional panel data set, validating multiple scientific facts.

Key words and phrases: B-spline, Electroencephalogram, moving average, oracle efficiency, simultaneous confidence region.