Defining Probability Density for a Distribution of Random Functions

• Activity Number: 661
• Type: Topic Contributed
• Date/Time: Thursday, August 5, 2010 : 10:30 AM to 12:20 PM
• Sponsor: Section on Nonparametric Statistics
• Author(s): Aurore Delaigle*+ and Peter Hall
• Companies: The University of Melbourne and The University of Melbourne
• Keywords: Functional data analysis ; density ; principal components

Abstract

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, 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 meaningful surrogate for density, defined in terms of the densities of the distributions of principal components. This density approximation is estimable readily from data. It accurately represents, in a monotone way, key features of small-ball approximations to density. The densities of principal component scores also reveal interesting shape differences. The statistical implications of these results and properties are identified and discussed, and practical ramifications are illustrated in numerical work.