Open Access
December 2010 A reproducing kernel Hilbert space approach to functional linear regression
Ming Yuan, T. Tony Cai
Ann. Statist. 38(6): 3412-3444 (December 2010). DOI: 10.1214/09-AOS772


We study in this paper a smoothness regularization method for functional linear regression and provide a unified treatment for both the prediction and estimation problems. By developing a tool on simultaneous diagonalization of two positive definite kernels, we obtain shaper results on the minimax rates of convergence and show that smoothness regularized estimators achieve the optimal rates of convergence for both prediction and estimation under conditions weaker than those for the functional principal components based methods developed in the literature. Despite the generality of the method of regularization, we show that the procedure is easily implementable. Numerical results are obtained to illustrate the merits of the method and to demonstrate the theoretical developments.


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Ming Yuan. T. Tony Cai. "A reproducing kernel Hilbert space approach to functional linear regression." Ann. Statist. 38 (6) 3412 - 3444, December 2010.


Published: December 2010
First available in Project Euclid: 30 November 2010

zbMATH: 1204.62074
MathSciNet: MR2766857
Digital Object Identifier: 10.1214/09-AOS772

Primary: 62J05
Secondary: 62G20

Keywords: Covariance , eigenfunction , eigenvalue , Functional linear regression , minimax , optimal convergence rate , Principal Component Analysis , ‎reproducing kernel Hilbert ‎space , Sacks–Ylvisaker conditions , simultaneous diagonalization , slope function , Sobolev space

Rights: Copyright © 2010 Institute of Mathematical Statistics

Vol.38 • No. 6 • December 2010
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