ORACLE INEQUALITIES FOR SPARSE ADDITIVE QUANTILE REGRESSION IN REPRODUCING KERNEL HILBERT SPACE

• 成果类型: Article
• 署名作者: Lv, Shaogao; Lin, Huazhen; Lian, Heng; Huang, Jian
• 署名单位: Nanjing Audit University; Southwestern University of Finance & Economics - China; City University of Hong Kong; University of Iowa
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/17-AOS1567
• 发表日期: 2018
• 页码: 781-813
• 关键词: VARIABLE SELECTION; model selection; Optimal Rates; Lasso; estimators; shrinkage

摘要

This paper considers the estimation of the sparse additive quantile regression (SAQR) in high-dimensional settings. Given the nonsmooth nature of the quantile loss function and the nonparametric complexities of the component function estimation, it is challenging to analyze the theoretical properties of ultrahigh-dimensional SAQR. We propose a regularized learning approach with a two-fold Lasso-type regularization in a reproducing kernel Hilbert space (RKHS) for SAQR. We establish nonasymptotic oracle inequalities for the excess risk of the proposed estimator without any coherent conditions. If additional assumptions including an extension of the restricted eigenvalue condition are satisfied, the proposed method enjoys sharp oracle rates without the light tail requirement. In particular, the proposed estimator achieves the minimax lower bounds established for sparse additive mean regression. As a by-product, we also establish the concentration inequality for estimating the population mean when the general Lipschitz loss is involved. The practical effectiveness of the new method is demonstrated by competitive numerical results.