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Optimal rate for prediction when predictor and response are functions. (English) Zbl 1457.62397

Summary: In functional data analysis, linear prediction problems have been widely studied based on the functional linear regression model. However, restrictive condition is needed to ensure the existence of the coefficient function. In this paper, a general linear prediction model is considered on the framework of reproducing kernel Hilbert space, which includes both the functional linear regression model and the point impact model. We show that from the point view of prediction, this general model works as well even the coefficient function does not exist. Moreover, under mild conditions, the minimax optimal rate of convergence is established for the prediction under the integrated mean squared prediction error. In particular, the rate reduces to the existing result when the coefficient function exists.

MSC:

62R10 Functional data analysis
62M20 Inference from stochastic processes and prediction
62G05 Nonparametric estimation
62G20 Asymptotic properties of nonparametric inference
62C20 Minimax procedures in statistical decision theory
60G25 Prediction theory (aspects of stochastic processes)
46E22 Hilbert spaces with reproducing kernels (= (proper) functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces)

Software:

fda (R); RKHS
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References:

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