Estimating and depicting the structure of a distribution of random functions

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. Copyright Biometrika Trust 2002, Oxford University Press.