Surface Functional Models.

The aim of this paper is to develop a new framework of surface functional models (SFM) for surface functional data which contains repeated observations in two domains (typically, time-location). The primary problem of interest is to investigate the relationship between a response and the two domains, where the numbers of observations in both domains within a subject may be diverging. The SFMs are far beyond the multivariate functional models with two-dimensional predictor variables. Unprecedented complexity presented in the surface functional models, such as possibly distinctive sampling designs and the dependence between the two domains, makes our models more complex than the existing ones. We provide a comprehensive investigation of the asymptotic properties of the local linear estimator of the mean function based on a general weighting scheme, including equal weight (EW), direction-to-denseness weight (DDW) and subject-to-denseness weight (SDW), as special cases. Moreover, we can mathematically categorize the surface data into nine cases according to the sampling designs (sparse, dense, and ultra-dense) of both the domains, essentially based on the relative order of the number of observations in each domain to the sample size. We derive the specific asymptotic theories and optimal bandwidth orders in each of the nine sampling design cases under all the three weighting schemes. The three weighting schemes are compared theoretically and numerically. We also examine the finite-sample performance of the estimators through simulation studies and an autism study involving white-matter fiber skeletons.