Cai, T. TonyYuan, Ming2023-05-232023-05-232012-01-012016-07-14https://repository.upenn.edu/handle/20.500.14332/47903This article 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 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. This motivates us to propose an easily implementable data-driven roughness regularization predictor that is shown to attain the optimal rate of convergence adaptively without the need of knowing the covariance kernel. Simulation studies are carried out to illustrate the merits of the adaptive predictor and to demonstrate the theoretical results.This is an Accepted Manuscript of an article published by Taylor & Francis in Journal of the American Statistical Association on 08 Oct 2012, available online: http://wwww.tandfonline.com/10.1080/01621459.2012.716337.functional linear modelminimax rate of convergenceprincipal components analysisreproducing kernel Hilbert spacespectral decompositionStatistics and ProbabilityArticle