Asymptotic properties of a nonparametric conditional density estimator in the local linear estimation for functional data via a functional single-index model

Abstract

This paper deals with the conditional density estimator of a real response variable given a functional random variable (i.e., takes values in an infinite-dimensional space). Specifically, we focus on the functional index model, and this approach represents a good compromise between nonparametric and parametric models. Then we give under general conditions and when the variables are independent, the quadratic error and asymptotic normality of estimator by local linear method, based on the single-index structure. Finally, we complete these theoretical advances by some simulation studies showing both the practical result of the local linear method and the good behaviour for finite sample sizes of the estimator and of the Monte Carlo methods to create functional pseudo-confidence area.

Keywords:

• Mean squared error
• single functional index
• conditional density function
• nonparametric estimation
• local linear estimation
• asymptotic normality
• functional data

Acknowledgments

The authors are very grateful to the Editor and the anonymous reviewers for their comments which improved the quality of this paper. The authors wish to thank two anonymous referees for their helpful comments and suggestions, which greatly improved the quality of this paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).