  This study examines the optimal selections of bandwidth and semi-metric for a
functional partial linear model. Our proposed method begins by estimating the
unknown error density using a kernel density estimator of residuals, where the
regression function, consisting of parametric and nonparametric components, can
be estimated by functional principal component and functional Nadayara-Watson
estimators. The estimation accuracy of the regression function and error
density crucially depends on the optimal estimations of bandwidth and
semi-metric. A Bayesian method is utilized to simultaneously estimate the
bandwidths in the regression function and kernel error density by minimizing
the Kullback-Leibler divergence. For estimating the regression function and
error density, a series of simulation studies demonstrate that the functional
partial linear model gives improved estimation and forecast accuracies compared
with the functional principal component regression and functional nonparametric
regression. Using a spectroscopy dataset, the functional partial linear model
yields better forecast accuracy than some commonly used functional regression
models. As a by-product of the Bayesian method, a pointwise prediction interval
can be obtained, and marginal likelihood can be used to select the optimal
semi-metric.
