Statistical Inference for Functional Data over Multi-dimensional Domain

Abstract

This work develops inference tools for the mean function of functional data over a multi-dimensional domain. Tensor product spline is used to recover individual trajectories, leading to a two-step mean estimator that is oracally efficient, meaning that it is asymptotically indistinguishable from the infeasible estimator using unobservable trajectories. A data-driven SCR with preset asymptotic coverage and uniformly adaptive width of order $n^{- 1/2}$ is established, supported by consistent estimates of covariance function as well as quantile interval of the maximal deviation process. The asymptotic theory extends to two-samples without extra difficulty. Extensive Monte Carlo experiments corroborate the theory, and a satellite ocean data collected by CMEMS illustrates how the proposed SCR is used.

(Dissertation work of Qirui Hu)