On the Local Linear Modelization of the Conditional Distribution for Functional Data

In this paper, we investigate the problem of the local linear estimation of the cumulative distribution function of a real random variable Y conditioned by a functional variable X (valued in an infinite dimensional space). The almost-complete and the mean square consistencies, with rates, of the constructed estimator are stated. We precise that the exact expression involved in the leading terms of the mean squared error is given. We point out, also, that the accuracy of our asymptotic results leads to interesting perspectives from a practical point of view. Thus, we discuss the features of our functional local modeling and the applicability of our asymptotic result on some statistical problems such as the choice of the smoothing parameters and the determination of confidence intervals. Moreover, a simulation study has been conducted in order to highlight, on a finite sample, the superiority of our method to the standard kernel method, in the functional framework.