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A reproducing kernel Hilbert space approach to functional linear regression. (English) Zbl 1204.62074

Summary: We study 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.

MSC:

62G08 Nonparametric regression and quantile regression
62J05 Linear regression; mixed models
46N30 Applications of functional analysis in probability theory and statistics
62G20 Asymptotic properties of nonparametric inference
62H25 Factor analysis and principal components; correspondence analysis
65C60 Computational problems in statistics (MSC2010)

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