Minimax and Adaptive Prediction for Functional Linear Regression
成果类型：
Article
署名作者：
Cai, T. Tony; Yuan, Ming
署名单位：
University of Pennsylvania; University System of Georgia; Georgia Institute of Technology
刊物名称：
JOURNAL OF THE AMERICAN STATISTICAL ASSOCIATION
ISSN/ISSBN：
0162-1459
DOI：
10.1080/01621459.2012.716337
发表日期：
2012
页码：
1201-1216
关键词：
摘要：
This 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.