ADAPTIVE ESTIMATION UNDER SINGLE-INDEX CONSTRAINT IN A REGRESSION MODEL

• 成果类型: Article
• 署名作者: Lepski, Oleg; Serdyukova, Nora
• 署名单位: Aix-Marseille Universite; Universidad de Concepcion
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/13-AOS1152
• 发表日期: 2014
• 页码: 1-28
• 关键词: random design; Nonlinear estimation; adaptation

摘要

The problem of adaptive multivariate function estimation in the single-index regression model with random design and weak assumptions on the noise is investigated. A novel estimation procedure that adapts simultaneously to the unknown index vector and the smoothness of the link function by selecting from a family of specific kernel estimators is proposed. We establish a pointwise oracle inequality which, in its turn, is used to judge the quality of estimating the entire function (global oracle inequality). Both the results are applied to the problems of pointwise and global adaptive estimation over a collection of Holder and Nikol'skii functional classes, respectively.