Kernel Conditional Density Estimation When the Regressor is Valued in a Semi-Metric Space

This article deals with the conditional density estimation when the explanatory variable is functional. In fact, nonparametric kernel type estimator of the conditional density has been recently introduced when the regressor is valued in a semi-metric space. This estimator depends on a smoothing parameter which controls its behavior. Thus, we aim to construct and study the asymptotic properties of a data-driven criterion for choosing automatically and optimally this smoothing parameter. This criterion can be formulated in terms of a functional version of cross-validation ideas. Under mild assumptions on the unknown conditional density, it is proved that this rule is asymptotically optimal. A simulation study and an application on real data are carried out to illustrate, for finite samples, the behavior of our method. Finally, we mention that our results can also be considered as novel in the finite dimensional setting and several other open questions are raised in this article.