Abstract: This study develops a simple method for constructing confidence bands centered at a principal component analysis (PCA)-based estimator of the slope function in a functional linear regression model with a scalar response variable and a functional predictor variable. A PCA-based estimator is a series estimator with estimated basis functions; thus, constructing these valid confidence bands is a nontrivial challenge. We propose a confidence band that covers most of the slope function with a prespecified probability (level), and prove its asymptotic validity under suitable regularity conditions. To the best of our knowledge, this is the first study to derive that derives confidence bands with theoretical justifications for the PCA-based estimator. We also propose a practical method for choosing the cutoff level used in the PCA-based estimation, and conduct numerical studies to verify the finite-sample performance of the proposed bands. Finally, we apply our methodology to spectrometric data, and discuss extensions of our methodology to cases where additional vector-valued regressors are present.

Key words and phrases: Confidence band, functional linear regression, functional principal component analysis.