Simultaneous inference for the mean function based on dense functional data

A polynomial spline estimator is proposed for the mean function of dense functional data together with a simultaneous confidence band which is asymptotically correct. In addition, the spline estimator and its accompanying confidence band enjoy oracle efficiency in the sense that they are asymptotically the same as if all random trajectories are observed entirely and without errors. The confidence band is also extended to the difference of mean functions of two populations of functional data. Simulation experiments provide strong evidence that corroborates the asymptotic theory while computing is efficient. The confidence band procedure is illustrated by analysing the near-infrared spectroscopy data.