On the Central Limit Theorem for Conditional Density Estimator In the Single Functional Index Model

The main objective of this paper is to 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 sample of observations is considered as an independent random variables with identical distribution (i.i.d.) and are linked with a single functional index structure. First of all, a kernel type estimator for the conditional density function (cond-df) is introduced. Afterwards, the asymptotic properties are stated for a conditional density estimator when the observations are linked with a single-index structure from which we derive an central limit theorem (CLT) of the conditional density estimator to show the asymptotic normality of the kernel estimate of this model. As an application the conditional mode in functional single-index model is presented. As an application the conditional mode in functional single-index model is presented as well as the asymptotic (1 - xi) confidence interval of the conditional mode function is given for 0 < xi < 1. Simulation study is also presented to illustrate the validity and finite sample performance of the considered estimator.