  The notion of probability density for a random function is not as
straightforward as in finite-dimensional cases. While a probability density
function generally does not exist for functional data, we show that it is
possible to develop the notion of density when functional data are considered
in the space determined by the eigenfunctions of principal component analysis.
This leads to a transparent and meaningful surrogate for density defined in
terms of the average value of the logarithms of the densities of the
distributions of principal components for a given dimension. This density
approximation is estimable readily from data. It accurately represents, in a
monotone way, key features of small-ball approximations to density. Our results
on estimators of the densities of principal component scores are also of
independent interest; they reveal interesting shape differences that have not
previously been considered. The statistical implications of these results and
properties are identified and discussed, and practical ramifications are
illustrated in numerical work.
