  The data functions that are studied in the course of functional data analysis
are assembled from discrete data, and the level of smoothing that is used is
generally that which is appropriate for accurate approximation of the
conceptually smooth functions that were not actually observed. Existing
literature shows that this approach is effective, and even optimal, when using
functional data methods for prediction or hypothesis testing. However, in the
present paper we show that this approach is not effective in classification
problems. There a useful rule of thumb is that undersmoothing is often
desirable, but there are several surprising qualifications to that approach.
First, the effect of smoothing the training data can be more significant than
that of smoothing the new data set to be classified; second, undersmoothing is
not always the right approach, and in fact in some cases using a relatively
large bandwidth can be more effective; and third, these perverse results are
the consequence of very unusual properties of error rates, expressed as
functions of smoothing parameters. For example, the orders of magnitude of
optimal smoothing parameter choices depend on the signs and sizes of terms in
an expansion of error rate, and those signs and sizes can vary dramatically
from one setting to another, even for the same classifier.
