Abstract

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 esti-

mates 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.

Information

Preprint No.SS-2024-0344
Manuscript IDSS-2024-0344
Complete AuthorsQirui Hu, Lijian Yang
Corresponding AuthorsLijian Yang
Emailsyanglijian@tsinghua.edu.cn

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Acknowledgments

This work is partially supported by the National Natural Science Foundation of

China under award No. 12171269.

Supplementary Materials

This supplement provides tables and figures of in Section 4, additional simulations

and detailed proofs of the theoretical results with necessarily technical lemmas.

Supplementary materials are available for download.