Consistency of recursive nonparametric Kernel estimates for independent functional data

We propose a new nonparametric estimator of the conditional hazard function. To this end we define nonparametric estimators of the conditional cumulative distribution and the density functions of a scalar response variable $Y$ given a functional random variable $X.$ The conditional cumulative distribution, density and hazard functions for independent functional data are estimated nonparametrically. Our estimates are based on a recursive approach. We establish under appropriate conditions the almost sure and the quadratic average convergence rates of the resulting hazard rate estimator. Furthermore, a simulation study and an application to a real dataset illustrate our methodology.