A note on the conditional density estimate in the single functional index model

In this paper, we consider the estimation of the conditional density of a scalar response variable YY, given a Hilbertian random variable XX when the observations are linked with a single-index structure. We establish the pointwise and the uniform almost complete convergence (with the rate) of the kernel estimate of this model. As an application, we show how our result can be applied in the prediction problem via the conditional mode estimate. Finally, the estimation of the functional index via the pseudo-maximum likelihood method is also discussed but not tackled.