Consistency results of the M-regression function estimator for stationary continuous-time and ergodic data

This paper is devoted to the study of asymptotic properties of the kernel estimator of the robust regression function for stationary continuous-time and ergodic data. Such a dependence structure is an alternative to the strong mixing conditions usually assumed in functional time series analysis. More precisely, we consider the kernel type estimator of the robust regression function constructed from the stationary and continuous-time ergodic data (Xt,Yt) for 0 <= t <= T. Then, we establish the almost sure (with rate) pointwise convergence of this estimator. A simulation study was conducted in order to compare the performance of this method to the classical regression method.