Non parametric estimations of the conditional density and mode when the regressor and the response are curves

We develop new estimation results for the functional relationship between a regressor and a response which are functions indexed by time or by spatial locations. The regressor is assumed to belong to a semi-metric space (E,d) whereas the responses belong to a Hilbert space F. First, we build a double-kernel estimator of the conditional density function, via a Nadaraya-Watson method. Then, we deduce a conditional mode estimator as the value that maximizes the conditional density estimator. Then, we establish the strong uniform consistencies, with rates, of the two constructed estimators. In this context, we wished to set up these preliminary results which will certainly motivate several works on this same subject.