Open Access
April 2010 Defining probability density for a distribution of random functions
Aurore Delaigle, Peter Hall
Ann. Statist. 38(2): 1171-1193 (April 2010). DOI: 10.1214/09-AOS741


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.


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Aurore Delaigle. Peter Hall. "Defining probability density for a distribution of random functions." Ann. Statist. 38 (2) 1171 - 1193, April 2010.


Published: April 2010
First available in Project Euclid: 19 February 2010

zbMATH: 1183.62061
MathSciNet: MR2604709
Digital Object Identifier: 10.1214/09-AOS741

Primary: 62G05
Secondary: 62G07

Keywords: 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

Rights: Copyright © 2010 Institute of Mathematical Statistics

Vol.38 • No. 2 • April 2010
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