A reproducing kernel Hilbert space approach to functional linear regression

Abstract

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.

Published in at http://dx.doi.org/10.1214/09-AOS772 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (4)

• Prediction in functional linear regression
• Methodology and convergence rates for functional linear regression
• Smoothing splines estimators for functional linear regression
• Nonparametric estimation in functional linear models with second order stationary regressors

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• A divergence formula for regularization methods with an L2 constraint
• Optimal Prediction in an Additive Functional Model
• Hybrid Regularisation of Functional Linear Models
• Minimax Prediction for Functional Linear Regression with Functional Responses in Reproducing Kernel Hilbert Spaces