Simultaneous inference for covariance function of next-generation functional data

Next-generation functional data arise from many fields nowadays, such as imaging data or spatial data in medical research. The covariance function takes an important role in analysis, serving as the first step for principal component decomposition and dimension reduction. In our paper, we propose a bivariate spline estimator for the covariance function based on discretely observed, error prone raw next-generation functional data. Flexible bivariate splines over triangulations are employed to accommodate the irregular domains that frequently appear in such data. The proposed covariance estimator enjoys oracle property in the sense that it is as efficient as the infeasible estimator when all individual images are entirely observed without errors. Both local and uniform weak convergence are investigated under some mild conditions, with the added bonus of asymptotic simultaneous confidence regions (SCRs). Extensive simulation experiments yield appealing results that strongly corroborate the asymptotic theory. The proposed method is finally illustrated by analyzing the brain positron emission tomography data from the Alzheimer’s Disease Neuroimaging Initiative (ADNI) database.