Estimating some characteristics of the conditional distribution in nonparametric functional models

This paper deals with a scalar response conditioned by a functional random variable. The main goal is to estimate nonparametrically some characteristics of this conditional distribution. Kernel type estimators for the conditional cumulative distribution function and the successive derivatives of the conditional density are introduced. Asymptotic properties are stated for each of these estimates, and they are applied to the estimations of the conditional mode and conditional quantiles. Our asymptotic results highlightes the importance of the concentration properties on small balls of the probability measure of the underlying functional variable. So, a special section is devoted to show how our results behave in several situations when the functional variable is a continuous time process, with special attention to diffusion processes and Gaussian processes. Even if the main purpose of our paper is theoretical, an application to some chemiometrical data set coming from food industry is presented in a short final section. This example illustrates the easy implementation of our method as well as its good behaviour for finite sample sizes. © Springer 2006.