Abstract: 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.

Key words and phrases: Cumulative distribution function, functional principal component scores, goodness of fit tests, nonparametric smoothing, Simultaneous inference.