Defining probability density for a distribution of random functions. (English) Zbl 1183.62061

Delaigle, Aurore; Hall, Peter

Ann. Stat. 38, No. 2, 1171-1193 (2010).

Summary: 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.

MSC:

62G07	Density estimation
62H25	Factor analysis and principal components; correspondence analysis
62G05	Nonparametric estimation

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; Australian rainfall data

Software:

SiZer; fda (R); KernSmooth

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