Robust optimal estimation of the mean function from discretely sampled functional data
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Estimating the mean of functional data 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 optimal class of robust estimators for the estimation of the mean from discretely sampled functional data. The proposed method is based on M-type smoothing spline estimation with repeated measurements and is suitable for densely observed trajectories as well as for sparsely observed trajectories that are subject to measurement error. Our analysis clearly delineates the role of the sampling frequency in the determination of the asymptotic properties of the M-type smoothing spline estimators: for commonly observed trajectories, the sampling frequency dominates the error when it is small but ceases to be important when it is large. On the other hand, for independently observed trajectories the sampling frequency plays a more limited role as the asymptotic error is jointly determined by the sampling frequency and the sample size. 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 that contains outlying observations.
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