Simultaneous inference for the mean function based on dense functional data

Abstract

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.

Keywords:

• B-spline
• confidence band
• functional data
• Karhunen–Loève L ² representation
• oracle efficiency

AMS Subject Classifications :

• Primary: 62M10
• Secondary: 62G08

Acknowledgements

This work has been supported in part by NSF awards DMS 0706518, 1007594, NCI/NIH K-award, 1K01 CA131259, a Dissertation Continuation Fellowship from Michigan State University, and funding from the Jiangsu Specially-Appointed Professor Programme, Jiangsu Province, China. The helpful comments by two referees and the Associate Editor have led to significant improvement of the paper.