A REPRODUCING KERNEL HILBERT SPACE APPROACH TO FUNCTIONAL LINEAR REGRESSION

• 成果类型: Article
• 署名作者: Yuan, Ming; Cai, T. Tony
• 署名单位: University System of Georgia; Georgia Institute of Technology; University of Pennsylvania
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/09-AOS772
• 发表日期: 2010
• 页码: 3412-3444
• 关键词: penalized likelihood; designs; CONVERGENCE; estimators; models; rates

摘要

We study in this paper 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.