Abstract:
There has been substantial
recent work on methods for estimating the slope function
in linear regression for functional data analysis.
However, as in the case of more conventional,
finite-dimensional regression, much of the practical
interest in the slope centres on its application for
the purpose of prediction, rather than on its
significance in its own right. We show that the
problems of slope-function estimation, and of prediction
from an estimator of the slope function, have very
different characteristics. While the former is
intrinsically nonparametric, the latter can be either
nonparametric or semiparametric. In particular, the
optimal mean-square convergence rate of predictors is
n^{-1/2},
where n denotes sample size, if the predictand
is a sufficiently smooth function. In other cases,
convergence occurs at a polynomial rate that is strictly
slower than n^{-1/2}.
At the boundary between these two
regimes, the mean-square convergence rate is less than
n^{-1/2} by only a
logarithmic factor. More generally,
the rate of convergence of the predicted value of the
mean response in the regression model, given a
particular value of the explanatory variable, is
determined by a subtle interaction among the smoothness
of the predictand, of the slope function in the model,
and of the autocovariance function for the distribution
of explanatory variables.