Simultaneous inference and uniform test for eigensystems of functional data

Abstract

The asymptotically correct confidence interval (CI) and simultaneous confidence band (SCB) of any individual eigenvalue and eigenfunction are constructed under dense functional data through B-spline smoothing. Besides, uniform inference procedures for eigensystems with a diverging number of components are novelly developed. The proposed estimators for functional eigensystems employ “oracle” efficiency up to order n, which means they are asymptotically indistinguishable from the estimators conducted by completely observed trajectories, and enjoy computational efficiency with much more convenient spectrum decomposition forms. Furthermore, an extension to two-sample problems is also investigated. Numerical simulation results strongly corroborate the asymptotic theory. Real data analysis for ElectroEncephalogram (EEG) data illustrates applicability of the developed methods.