On the Nonparametric Conditional Density and Mode Estimates in the Single Functional Index Model with Strongly Mixing Data

Abstract

This study focuses on the nonparametric estimation of the conditional density of a scalar response variable given a random variable taking values in separable Hilbert space. We establish under general conditions the almost complete convergence rates of the conditional density estimator under α-mixing dependence, based on the single-index structure. We also demonstrate the impact of this functional parameter on the mode estimation. Finally, the estimation of the functional index via the pseudo-maximum likelihood method is discussed but not tackled.