Strong Uniform Consistency Rates of Conditional Quantiles with Functional Variables in the Functional Single-Index Model

Abstract

This paper deals with a scalar response conditioned by a functional random variable. The main goal is to estimate nonparametrically the quantiles of such a conditional distribution when the sample is considered as an i.i.d sequence. Firstly, a kernel type estimator for the conditional cumulative distribution function (cond-cdf) is introduced. Afterwards, we derive an estimation of the quantiles by inverting this estimated cond-cdf and asymptotic properties are stated when the observations are linked with a single-index structure. We establish the pointwise almost complete convergence and the uniform almost complete convergence (with the rate) of the kernel estimate of this model. The functional conditional quantile approach can be used both to forecast and to build confidence prediction bands.