Abstract: We propose a test for the hypothesis that the standardized functional principal components (FPCs) of functional data are equal to a given set of orthonormal bases (e.g., the Fourier basis). Using estimates of individual trajectories that satisfy certain approximation conditions, we construct a chi-square-type statistic, and show that it is oracally efficient under the null hypothesis, in the sense that its limiting distribution is the same as that of an infeasible statistic using all trajectories, known as the oracle. The null limiting distribution is an infinite Gaussian quadratic form, and we obtain a consistent estimator of its quantile. A test statistic based on the chi-squared-type statistic and the approximate quantile of the Gaussian quadratic form is shown to be both of the nominal asymptotic significance level and asymptotically correct. It is further shown that B-spline trajectory estimates meet the required approximation conditions. Simulation studies demonstrate the superior finite-sample performance of the proposed testing procedure. Using electroencephalogram (EEG) data, the proposed procedure confirms an interesting discovery that the centered EEG data are generated from a small number of elements of the standard Fourier basis.

Key words and phrases: B-spline, ElectroEncephalogram, functional principal components, Gaussian quadratic form, oracle efficiency