ANALYSIS OF VARIANCE, COEFFICIENT OF DETERMINATION AND F-TEST FOR LOCAL POLYNOMIAL REGRESSION

• 成果类型: Article
• 署名作者: Huang, Li-Shan; Chen, Jianwei
• 署名单位: University of Rochester; California State University System; San Diego State University
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/07-AOS531
• 发表日期: 2008
• 页码: 2085-2109
• 关键词: Nonparametric regression; correlation curves; checking; weak

摘要

This paper provides ANOVA inference for nonparametric local polynomial regression (LPR) in analogy with ANOVA tools for the classical linear regression model. A surprisingly simple and exact local ANOVA decomposition is established, and a local R-squared quantity is defined to measure the proportion of local variation explained by fitting LPR. A global ANOVA decomposition is obtained by integrating local counterparts, and a global R-squared and a symmetric projection matrix are defined. We show that the proposed projection matrix is asymptotically idempotent and asymptotically orthogonal to its complement, naturally leading to an F-test for testing for no effect. A by-product result is that the asymptotic bias of the projected response based on local linear regression is of quartic order of the bandwidth. Numerical results illustrate the behaviors of the proposed R-squared and F-test. The ANOVA methodology is also extended to varying coefficient models.