The aim of this paper is to establish a nonparametric estimate of some characteristics of the conditional distribution. Kernel type estimators for the conditional cumulative distribution function and for the successive derivatives of the conditional density of a scalar response variable Y given a Hilbertian random variable X are introduced when the observations are linked with a single-index structure. We establish the pointwise almost complete convergence and the uniform almost complete convergence (with rate) of the kernel estimator of this model. Asymptotic properties are stated for each of these estimators, and they are applied to the estimation of the conditional mode and conditional quantiles.

Abstract. The maximum of the conditional hazard function is a parameter of great importance in statistics, in particular in seismicity studies, because it constitutes the maximum risk of occurrence of an earthquake in a given interval of time. Using the kernel nonparametric estimates based on convolution kernel techniques of the first derivative of the conditional hazard function, we establish the asymptotic behavior of a hazard rate in the presence of a functional explanatory variable and asymptotic normality of the maximum value in the case of a strong mixing process.Résumé. Le maximum ou encore le pointà haut risque d'une fonction de risque conditionnel est un paramètre d'un grand intérêt en statistique, notamment dans l'analyse de risque séismique, car il constitue le risque maximal de survenance d'un tremblement de terre dans un intervalle de temps donné. Au moyen d'estimations non paramétriques basés sur les techniques de noyau de convolution de la première dérivée de la fonction de hasard conditionnel, nousétablissons le comportement asymptotique d'un taux de hasard d'une variable explicative fonctionnelle ainsi que la normalité asymptotique de la valeur maximale pour un processus mélangeant.