Revisiting the estimation of the error density in functional autoregressive models

In this paper, we study some asymptotic properties of a new estimator of the probability density function of the driven noise in a nonparametric functional autoregressive model. This density estimator, based on the kernel method, does not require the estimation of the residuals of the model (as in the usual plug-in estimator). We prove its pointwise consistency with rate and establish a multivariate central limit theorem, without any assumption on the noise distribution tail. Theoretical results are illustrated with some simulation experiments. We finally propose a goodness-of-fit test of the error distribution, built from a normalized sum of the quadratic deviation between the true density and its estimator evaluated on a finite set of distinct points.