Oracle-Efficient Confidence Envelopes for Covariance Functions in Dense Functional Data

We consider nonparametric estimation of the covariance function for dense functional data using computationally efficient tensor product B-splines. We develop both local and global asymptotic distributions for the proposed estimator, and show that our estimator is as efficient as an “oracle” estimator where the true mean function is known. Simultaneous confidence envelopes are developed based on asymptotic theory to quantify the variability in the covariance estimator and to make global inferences on the true covariance. Monte Carlo simulation experiments provide strong evidence that corroborates the asymptotic theory. Examples of near infrared spectroscopy data and speech recognition data are provided to illustrate the proposed method.