Strong uniform consistency rates of some characteristics of the conditional distribution estimator in the functional single-index model

Abstract

The aim of this paper is to establish a nonparametric estimate of some characteristics of the conditional distribution. Kernel type estimators for the conditional cumulative distribution function and for the successive derivatives of the conditional density of a scalar response variable Y given a Hilbertian random variable X are introduced when the observations are linked with a single-index structure. We establish the pointwise almost complete convergence and the uniform almost complete convergence (with rate) of the kernel estimator of this model. Asymptotic properties are stated for each of these estimators, and they are applied to the estimation of the conditional mode and conditional quantiles.

Asymptotic Results of a Nonparametric Conditional Quantile Estimator in the Single Functional Index Modeling under Random Censorship

“…1. These results extends Theorem 3 or Theorem 4 given in Bouchentouf et al [6] to the mixing case. The effect of covariance structure for dependence case on the convergence rate is reflected in the last term.…”

“…The effect of covariance structure for dependence case on the convergence rate is reflected in the last term. Specially, if the functional single-index is fixed, it is easy to prove the following corollary that are similar the one given in Bouchentouf et al [6].…”

“…• (H1)-(H4) and (H7) are standard assumptions for the distribution conditional estimation in single functional index model, which have been adopted by Bouchentouf et al [6] for i.i.d case.…”

“…Proof. [Proof of Lemma 7] Similar to the proof of Lemma 4.4 in Bouchentouf et al [6], it can be completed easily. Here we omit its proof.…”

Asymptotic normality of conditional density and conditional mode in the functional single index model

“…These models attracted the attention of many researchers, such as Aït-Saidi, Ferraty and Kassa (2005). Bouchentouf, Djebbouri, Rabhi, and Sabri (2014) established a nonparametric estimation of some characteristics of the conditional cumulative distribution function and the successive derivatives of the conditional density of a scalar response variable Y given a Hilbertian random variable X when the observations are linked with a single-index structure. Attaoui, Laksaci, and Ould-Saïd (2011) studied the functional single-index model via its conditional density kernel estimator, and established its pointwise and uniform almost complete convergence rates, and their results were extended to the dependent case by Attaoui (2014).…”