Simultaneous inference for the mean of repeated functional data

Motivated by recent works studying the longitudinal diffusion tensor imaging (DTI) studies, we develop a novel procedure to construct simultaneous confidence bands for mean functions of repeatedly observed functional data. A fully nonparametric method is proposed to estimate the mean function and variance-covariance function of the repeated trajectories via polynomial spline smoothing. The proposed confidence bands are shown to be asymptotically correct by taking into account the correlation of trajectories within subjects. The procedure is also extended to the two-sample case in which we focus on comparing the mean functions from two populations of functional data. We show the finite-sample properties of the proposed confidence bands by simulation studies, and compare the performance of our approach with the “naive” method that assumes the independence within the repeatedly observed trajectories. The proposed method is applied to the DTI study.