Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Defining probability density for a distribution of random functions   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.

24
The categories of this paper are: dg-ga
28
The categories of this paper are: alg-geom
12
The categories of this paper are: nucl-th
9
The categories of this paper are: nlin
3
The categories of this paper are: physics
19
The categories of this paper are: eess
33
The categories of this paper are: plasm-ph
21
The categories of this paper are: adap-org
20
The categories of this paper are: acc-phys
14
The categories of this paper are: math-ph
18
The categories of this paper are: econ
11
The categories of this paper are: hep-lat
31
The categories of this paper are: ao-sci
5
The categories of this paper are: gr-qc
22
The categories of this paper are: chao-dyn
29
The categories of this paper are: funct-an