Conditional Density Estimation in the Single Functional Index Model for α-Mixing Functional Data

In this article, we investigate the nonparametric estimation of the conditional density of a scalar response variable Y, given the explanatory variable X taking value in a Hilbert space when the observations are linked with a single index structure. The goal of this article is to present the asymptotic results such as pointwise almost complete consistency and the uniform almost complete convergence of the kernel estimation with rate for the conditional density in the setting of the -mixing functional data, which extend the i.i.d case in Attaoui et al. (2011) to the dependence setting. As an application, the convergence rate of the kernel estimation for the conditional mode is also obtained.