Equipe Probabilités et statistiques du 28 juin 2018
This paper aims to study the asymptotic properties of the conditional variance estimator in a nonlinear heteroscedastic functional regression model with martingale difference errors. A kernel-type estimator of the conditional variance is introduced when the response is a real-valued random variable and the covariate takes values in an infinite dimensional space endowed with a semi-metric. Under stationarity and ergodicity assumptions, a uniform almost sure consistency rate as well as the asymptotic distribution of the estimator are established. To build confidence intervals for the conditional variance, two approaches are discussed. The first one is based on the normal approximation approach and the second applies empirical likelihood method. We stress on the fact that errors are assumed to form a martingale difference and may depend on the covariate. Moreover, our results hold under a general dependency structure (ergodicity) and without assuming any mixing conditions which allow to cover a larger class of dependent processes. Two simulation studies are carried out to show the performance of the proposed estimator and to compare the two methods in building confidence intervals. An application to volatility prediction of the daily peak electricity demand in France, using the previous intraday load curve, is also provided.