Robust optimal estimation of the mean function from discretely sampled functional data

Estimating location is a central problem in functional data analysis, yet most current estimation procedures either unrealistically assume completely observed trajectories or lack robustness with respect to the many kinds of anomalies one can encounter in the functional setting. To remedy these deficiencies we introduce the first class of optimal robust location estimators based on discretely sampled functional data. The proposed method is based on M-type smoothing spline estimation with repeated measurements and is suitable for both commonly and independently observed trajectories that are subject to measurement error. We show that for commonly observed trajectories the proposed family of estimators is minimax rate optimal while for independently observed trajectories it achieves the optimal nonparametric rate regardless of whether the trajectories are densely or sparsely sampled. We illustrate the excellent performance of the proposed family of estimators relative to existing methods in a Monte-Carlo study and a real-data example.