Nonparametric estimation of the conditional mode when the regressor is functional.

We study a kernel estimator of the conditional mode of a scalar response variable Y given a random variable X taking values in a semi-metric space. We establish the consistency in L-p norm of the estimator. The asymptotic results are closely related to the concentration properties on small balls of the probability measure of the underlying explanatory variable and the regularity of the conditional density. Our conditions and results unify both cases of finite and infinite dimensional regressors.