  This paper develops a simple method to construct confidence bands, centered
at a principal component analysis (PCA) based estimator, for the slope function
in a functional linear regression model with a scalar response variable and a
functional predictor variable. The PCA-based estimator is a series estimator
with estimated basis functions, and so construction of valid confidence bands
for it is a non-trivial challenge. We propose a confidence band that aims at
covering the slope function at "most" of points with a prespecified probability
(level), and prove its asymptotic validity under suitable regularity
conditions. Importantly, this is the first paper that derives confidence bands
having theoretical justifications for the PCA-based estimator. We also propose
a practical method to choose the cut-off level used in PCA-based estimation,
and conduct numerical studies to verify the finite sample performance of the
proposed confidence band. Finally, we apply our methodology to spectrometric
data, and discuss extensions of our methodology to cases where additional
vector-valued regressors are present.
