Simultaneous Inference for the Distribution of Functional Principal Component Scores

This paper introduces a novel methodology for simultaneous inference of the cumulative distribution function (CDF) of functional principal component (FPC) scores.We establish a general framework for estimating the CDF, including both nonsmooth and smooth estimators, and demonstrate their asymptotic equivalence.For dense functional data, we employ nonparametric pre-smoothing, ensuring oracle properties that make our estimators equivalent to those from fully observed trajectories.We recommend B-spline smoothing for its computational efficiency.Additionally, we derive theoretical properties to construct simultaneous confidence bands (SCBs) and develop new testing procedures for the distribution of FPC scores.These procedures, including Kolmogorov-Smirnov and Cramér-von Mises tests, can handle a diverging number of components and are particularly effective for testing the normality of functional data, a common assumption in literature and practice.Our methodology is supported by extensive numerical simulations and applied to well-known functional datasets and Electroencephalogram (EEG) data.