The Annals of Statistics
- Ann. Statist.
- Volume 38, Number 2 (2010), 1171-1193.
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.
Ann. Statist., Volume 38, Number 2 (2010), 1171-1193.
First available in Project Euclid: 19 February 2010
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Density estimation dimension eigenfunction eigenvalue functional data analysis kernel methods log-density estimation nonparametric statistics principal components analysis probability density function resolution level scale space
Delaigle, Aurore; Hall, Peter. Defining probability density for a distribution of random functions. Ann. Statist. 38 (2010), no. 2, 1171--1193. doi:10.1214/09-AOS741. https://projecteuclid.org/euclid.aos/1266586626