Statistical Inference for Functional Data over Multi Dimensional Domain

This work develops inference tools for the mean function of functional data over a multi-dimensional domain.A two-step mean estimator based on tensor product spline estimates of individual trajectories is shown oracally efficient, i.e., it is asymptotically indistinguishable from the infeasible estimator using unobservable trajectories.Consistent estimates of covariance function as well as exact quantile of the limiting maximal deviation are obtained by innovative use of results on sharp comparison of Gaussian extreme distributions and quantiles, leading to asymptotic coverage and order n -1/2 uniformly adaptive width of data-driven simultaneous confidence regions (SCRs).Also formulated are one-sided SCRs that can be used for testing against uniform upper and lower bound of the mean function.Extensive Monte Carlo experiments corroborate the theory, and a satellite ocean dataset collected by Copernicus Marine Environment Monitoring Service (CMEMS) illustrates how the proposed SCR is used.