  Recent technological developments have enabled us to collect complex and
high-dimensional data in many scientific fields, such as population health,
meteorology, econometrics, geology, and psychology. It is common to encounter
such datasets collected repeatedly over a continuum. Functional data, whose
sample elements are functions in the graphical forms of curves, images, and
shapes, characterize these data types. Functional data analysis techniques
reduce the complex structure of these data and focus on the dependences within
and (possibly) between the curves. A common research question is to investigate
the relationships in regression models that involve at least one functional
variable. However, the performance of functional regression models depends on
several factors, such as the smoothing technique, the number of basis
functions, and the estimation method. This paper provides a selective
comparison for function-on-function regression models where both the response
and predictor(s) are functions, to determine the optimal choice of basis
function from a set of model evaluation criteria. We also propose a bootstrap
method to construct a confidence interval for the response function. The
numerical comparisons are implemented through Monte Carlo simulations and two
real data examples.
