Nonparametric Estimation for Additive Concurrent Regression Models

Consider an additive functional regression model where a one-dimensional response process Y(t) and a Kdimensional explanatory random process Xj(t), j = 1,.. ., K, are observed fort E [0, T], with fixed T. Examples of such explanatory processes are continuous or inhomogeneous counting processes. The linear coefficients of the model are K unknown deterministic functions t -> bj(t), j = 1,.. ., K, t E [0, T]. From N independent trajectories, we build a nonparametric least-squares estimator ( b1, ... , bK) of (b1, ... , bK), where each bj, 1 <= j <= K, is given by its expansion on a finite-dimensional space. We prove a bound on the mean-square risk of the estimator, from which rates of convergence are obtained and are established to be optimal. An adaptive procedure, achieving simultaneous and anisotropic selection of each space dimension, is then tailored and an oracle risk bound is proved. The procedure is studied numerically and implemented on a real dataset of electric consumption.