Simultaneous Confidence Band for Stationary Covariance Function of Dense Functional Data

TL;DR

This paper develops a method to construct simultaneous confidence bands for the stationary covariance function of dense functional data, using a two-stage spline-based estimation process with proven asymptotic properties.

Contribution

It introduces a novel two-stage spline-based estimation procedure for the covariance function and constructs asymptotically correct simultaneous confidence bands.

Findings

The estimator is as efficient as the oracle estimator when all trajectories are known.

The asymptotic confidence band has correct coverage probabilities.

Simulation and real data examples demonstrate the method's effectiveness.

Abstract

Inference via simultaneous confidence band is studied for stationary covariance function of dense functional data. A two-stage estimation procedure is proposed based on spline approximation, the first stage involving estimation of all the individual trajectories and the second stage involving estimation of the covariance function through smoothing the empirical covariance function. The proposed covariance estimator is smooth and as efficient as the oracle estimator when all individual trajectories are known. An asymptotic simultaneous confidence band (SCB) is developed for the true covariance function, and the coverage probabilities are shown to be asymptotically correct. Simulation experiments are conducted on the numerical performance of the proposed estimator and SCB. The proposed method is also illustrated by two real data examples.