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

Huang, Li-Shan; Chen, Jianwei

Ann. Stat. 36, No. 5, 2085-2109 (2008).

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.

MSC:

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

Keywords:

bandwidth; nonparametric regression; projection matrix; R-squared; smoothing splines; varying coefficient models; model checking

Software:

KernSmooth; SemiPar

References:

[1]	Azzalini, A., Bowman, A. W. and Hardle, W. (1989). On the use of nonparametric regression for model checking. Biometrika 76 1-11. JSTOR: · Zbl 0663.62096
[2]	Azzalini, A. and Bowman, A. W. (1993). On the use of nonparametric regression for checking linear relationships. J. Roy. Statist. Soc. Ser. B 55 549-557. JSTOR: · Zbl 0800.62222
[3]	Bjerve, S. and Doksum, K. (1993). Correlation curves: Measures of association as functions of covariate values. Ann. Statist. 21 890-902. · Zbl 0817.62025
[4]	Doksum, K., Blyth, S., Bradlow, E., Meng, X.-L. and Zhao, H. (1994). Correlation curves as local measures of variance explained by regression. J. Amer. Statist. Assoc. 89 571-572. JSTOR: · Zbl 0803.62035
[5]	Doksum, K. and Samarov, A. (1995). Global functionals and a measure of the explanatory power of covariates in regression. Ann. Statist. 23 1443-1473. · Zbl 0843.62045
[6]	Doksum, K. and Froda, S. M. (2000). Neighborhood correlation. J. Statist. Plann. Inference 91 267-294. · Zbl 0970.62019
[7]	Draper, N. R. and Smith, H. (1981). Applied Regression Analysis , 2nd ed. Wiley, New York. · Zbl 0548.62046
[8]	Eubank, R. L. (1999). Nonparametric Regression and Spline Smoothing , 2nd ed. Dekker, New York. · Zbl 0936.62044
[9]	Fan, J. and Gijbels, I. (1996). Local Polynomial Modelling and Its Applications . Chapman and Hall, London. · Zbl 0873.62037
[10]	Fan, J., Zhang, C. and Zhang, J. (2001). Generalized likelihood ratio statistics and Wilks phenomenon. Ann. Statist. 29 153-193. · Zbl 1029.62042
[11]	Fan, J. and Zhang, W. (1999). Statistical estimation in varying coefficient models. Ann. Statist. 27 1491-1518. · Zbl 0977.62039
[12]	Gijbels, I. and Rousson, V. (2001). A nonparametric least-squares test for checking a polynomial relationship. Statist. Probab. Lett. 51 253-261. · Zbl 0965.62038
[13]	Hastie, T. J. and Tibshirani, R. J. (1990). Generalized Additive Models . Chapman and Hall, London. · Zbl 0747.62061
[14]	Hastie, T. J. and Tibshirani, R. J. (1993). Varying-coefficient models. J. Roy. Statist. Soc. Ser. B 55 757-796. JSTOR: · Zbl 0796.62060
[15]	Hoover, D. R., Rice, J. A., Wu, C. O. and Yang, L. P. (1998). Nonparametric smoothing estimates of time-varying coefficient models with longitudinal data. Biometrika 85 809-822. JSTOR: · Zbl 0921.62045
[16]	Huang, L.-S. and Su, H. (2006). Nonparametric F -tests for nested global and local polynomial models. Technical Report 2006-08, Dept. of Biostatistics and Computational Biology, Univ. Rochester. · Zbl 1153.62036
[17]	Lehmann, E. L. and Casella, G. (1998). Theory of Point Estimation . Springer, New York. · Zbl 0916.62017
[18]	Mack, Y. P. and Silverman, B. W. (1982). Weak and strong uniform consistency of kernel regression and density estimation. Probab. Theory Related Fields 61 405-415. · Zbl 0495.62046
[19]	Mammen, E., Linton, O. B. and Nielsen, J. P. (1999). The existence and asymptotic properties of a backfitting projection algorithm under weak conditions. Ann. Statist. 27 1443-1490. · Zbl 0986.62028
[20]	Nielsen, J. P. and Sperlich, S. (2005). Smooth backfitting in practice. J. Roy. Statist. Soc. Ser. B 67 43-61. JSTOR: · Zbl 1060.62048
[21]	Qiu, P. (2003). A jump-preserving curve fitting procedure based on local piecewise-linear kernel estimation. J. Nonparametr. Statist. 15 437-453. · Zbl 1054.62047
[22]	Ramil-Novo, L. A. and González-Manteiga, W. (2000). F -tests and regression analysis of variance based on smoothing spline estimators. Statist. Sinica 10 819-837. · Zbl 0952.62037
[23]	Ruppert, D., Wand, M. P. and Carroll, R. J. (2003). Semiparametric Regression . Cambridge Univ. Press, London. · Zbl 1038.62042
[24]	Silverman, B. W. (1986). Density Estimation for Statistics and Data Analysis . Chapman and Hall, London. · Zbl 0617.62042
[25]	Simonoff, J. S. (1996). Smoothing Methods in Statistics . Springer, New York. · Zbl 0859.62035
[26]	Wand, M. P. and Jones, M. C. (1995). Kernel Smoothing . Chapman and Hall, London. · Zbl 0854.62043
[27]	Zhang, C. (2003). Calibrating the degrees of freedom for automatic data smoothing and effective curve checking. J. Amer. Statist. Assoc. 98 609-628. · Zbl 1040.62027