Minimax and Adaptive Prediction for Functional Linear Regression
Tony Cai and Ming Yuan
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Abstract:
This paper considers minimax and adaptive prediction with functional predictors in the framework of functional linear model and reproducing kernel Hilbert space. Minimax rate of convergence for the excess prediction risk is established. It is shown that the optimal rate is determined jointly by the reproducing kernel and the covariance function/kernel. In particular, the alignment of these two kernels can significantly affect the difficulty of the prediction problem. In contrast, the existing literature has so far focused only on the setting where the two kernels are nearly perfectly aligned. A data-driven roughness regularization predictor is introduced and is shown to attain the optimal rate of convergence adaptively without the need of knowing the covariance function. The procedure is easy to implement. Simulation studies are carried out to illustrate the merits of the adaptive predictor and to demonstrate the theoretical results.
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