Statistical Inference for Mean Function of Longitudinal Imaging Data over Complicated Domains

We propose a novel procedure for estimating the mean function of longitudinal imaging data with inherent spatial and temporal correlation. We depict the dependence between temporally ordered images using a functional moving average, and use flexible bivariate splines over triangulations to handle the irregular domain of images which is common in imaging studies. We establish both the global and the local asymptotic properties of the bivariate spline estimator for the mean function, with simultaneous confidence corridors (SCCs) as a theoretical byproduct. Under some mild conditions, the proposed estimator and its accompanying SCCs are shown to be consistent and oracle efficient, as though all images were entirely observed without errors. We use Monte Carlo simulation experiments to demonstrate the finite-sample performance of the proposed method, the results of which strongly corroborate the asymptotic theory. The proposed method is further illustrated by analyzing two seawater potential temperature data sets.