Analysis of variance, coefficient of determination and \(F\)-test for local polynomial regression. (English) Zbl 1148.62055

Summary: 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 effects. 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.


62J10 Analysis of variance and covariance (ANOVA)
62G08 Nonparametric regression and quantile regression
62H12 Estimation in multivariate analysis
65C60 Computational problems in statistics (MSC2010)


KernSmooth; SemiPar
Full Text: DOI arXiv


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