From Sparse to Dense Functional Data: Phase Transitions from a Simultaneous Inference Perspective

TL;DR

This paper develops a unified framework for simultaneous inference of mean functions in functional data analysis, covering sparse to dense data regimes, and explores phase transitions through Gaussian approximation and variance decomposition.

Contribution

It introduces a unified Gaussian approximation approach for confidence bands and analyzes phase transitions in asymptotic variance for different estimators.

Findings

The method effectively constructs simultaneous confidence bands across data sparsity levels.

Phase transition conditions are characterized via asymptotic variance decomposition.

Simulation and real data applications validate the theoretical results.

Abstract

We aim to develop simultaneous inference tools for the mean function of functional data from sparse to dense. First, we derive a unified Gaussian approximation to construct simultaneous confidence bands of mean functions based on the B-spline estimator. Then, we investigate the conditions of phase transitions by decomposing the asymptotic variance of the approximated Gaussian process. As an extension, we also consider the orthogonal series estimator and show the corresponding conditions of phase transitions. Extensive simulation results strongly corroborate the theoretical results, and also illustrate the variation of the asymptotic distribution via the asymptotic variance decomposition we obtain. The developed method is further applied to body fat data and traffic data.