Mathematics – Statistics Theory
Scientific paper
2008-10-27
Annals of Statistics 2008, Vol. 36, No. 5, 2085-2109
Mathematics
Statistics Theory
Published in at http://dx.doi.org/10.1214/07-AOS531 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Scientific paper
10.1214/07-AOS531
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.
Chen Jianwei
Huang Li-Shan
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