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Biometrika 2002 89(1):145-158; doi:10.1093/biomet/89.1.145
© 2002 by Biometrika Trust
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Estimating and depicting the structure of a distribution of random functions

Peter Hall1 and Nancy E.Heckman2

1 Centre for Mathematics and its Applications, Australian National University, Canberra, ACT 0200, Australia peter.hall{at}anu.edu.au 2 Department of Statistics, University of British Columbia, Vancouver BC V6T 1Z2, Canada heckman{at}stat.ubc.ca

We suggest a nonparametric approach to making inference about the structure of distributions in a potentially infinite-dimensional space, for example a function space, and displaying information about that structure.It is suggested that the simplest way of presenting the structure is through modes and density ascent lines, the latter being the projections into the sample space of the curves of steepest ascent up the surface of a functional-data density. Modes are always points in the sample space, and ascent lines are always one-parameter structures, even when the sample space is determined by an infinite number of parameters. They are therefore relatively easily depicted. Our methodology is based on a functional form of an iterative data-sharpening algorithm.

Key Words: Bandwidth; Cluster analysis; Functional data analysis; Gaussian process; Generalised Fourier expansion; Karhunen–Loève expansion; Kernel methods; Line of steepest ascent; Mode; Nonparametric density estimation; Tree diagram

Received July 2000. Revised April 2001


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