Abstract
This paper studies local M-estimation of the nonparametric components of additive models. A two-stage local M-estimation procedure is proposed for estimating the additive components and their derivatives. Under very mild conditions, the proposed estimators of each additive component and its derivative are jointly asymptotically normal and share the same asymptotic distributions as they would be if the other components were known. The established asymptotic results also hold for two particular local M-estimations: the local least squares and least absolute deviation estimations. However, for general two-stage local M-estimation with continuous and nonlinear ψ-functions, its implementation is time-consuming. To reduce the computational burden, one-step approximations to the two-stage local M-estimators are developed. The one-step estimators are shown to achieve the same efficiency as the fully iterative two-stage local M-estimators, which makes the two-stage local M-estimation more feasible in practice. The proposed estimators inherit the advantages and at the same time overcome the disadvantages of the local least-squares based smoothers. In addition, the practical implementation of the proposed estimation is considered in details. Simulations demonstrate the merits of the two-stage local M-estimation, and a real example illustrates the performance of the methodology.
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References
Friedman J H, Stuetzle W. Projection pursuit regression. J Amer Statist Assoc, 76: 817–823 (1981)
Stone C J. Additive regression and other nonparametric models. Ann Statist, 13: 689–705 (1985)
Stone C J. The dimensionality reduction principle for generalized additive models. Ann Statist, 14: 590–606 (1986)
Hastie T, Tibshirani R J. Generalized Additive Models. London: Chapman & Hall, 1990
Breiman L, Friedman J H. Estimating optimal transformations for multiple regression and correlation. J Amer Statist Assoc, 80: 580–619 (1985)
Buja A, Hastie T, Tibshirani R J. Linear smoothers and additive models. Ann Statist, 17: 453–510 (1989)
Opsomer J D, Ruppert D. Fitting a bivariate additive model by local polynomial regression. Ann Statist, 25: 186–211 (1997)
Mammen E, Linton O, Nielsen J P. The existence and asymptotic properties of backfitting projection algorithm under weak conditions. Ann Statist, 27: 1443–1490 (1999)
Opsomer J D. Asymptotic properties of backfitting estimator. J Multivariate Anal, 73: 166–179 (2000)
Tjøstheim D, Auestad B H. Nonparametric identification of nonlinear time series: Projections. J Amer Statist Assoc, 89: 1398–1409 (1994)
Linton O, Nielsen J P. A kernel method of estimating structured nonparametric regression based on marginal integration. Biometrika, 82: 93–100 (1995)
Chen R, Härdle W, Linton O, et al. Nonparametric estimation of additive separable regression models. In: Härdle W, Schimek M, eds. Statistical Theory and Computational Aspects of Smoothing. Heidelberg: Physica, 1996, 247–253
Linton O, Härdle W. Estimating additive regression models with known link function. Biometrika, 83: 529–540 (1996)
Fan J, Härdle W, Mammen E. Direct estimation of low-dimensional components in additive models. Ann Statist, 26: 943–971 (1998)
Stone C J. The use of polynomial splines and their tensor products in multivariate function estimation. Ann Statist, 22: 118–184 (1994)
Newey W K. Convergence rates and asymptotic normality for series estimators. J Multivariate Anal, 73: 147–168 (1997)
Horowitz J L, Mammen E. Nonparametric estimation of an additive model with a link function. Ann Statist, 32: 2412–2443 (2004)
Linton O. Estimating additive nonparametric models by partial L q norm: the curse of fractionality. Econometric Theory, 17: 1037–1050 (2001)
Fan J, Jiang J. Variable bandwidth and one-step local M-estimator. Sci China Ser A-Math, 43: 65–81 (2000)
Jiang J, Mack Y P. Robust local polynomial regression for dependent data. Statist Sinica, 11: 705–722 (2001)
Huber P J. Robust Statistics. New York: Wiley, 1981
He X, Shi P. Bivariate tensor-product B-splines in a partly linear model. J Multivariate Anal, 58: 162–181 (1996)
Doksum K, Koo J-Y. On spline estimators and prediction intervals in nonparametric regression. Comput Statist Data Anal, 35: 67–82 (2000)
Horowitz J L, Lee S. Nonparametric estimation of an additive quantile regression model. J Amer Statist Assoc, 100: 1238–1249 (2005)
De Boor C. A Practical Guilde to Splines. New York: Springer-Verlag, 1978
Harrison D, Rubinfeld D L. Hedonic housing prices and the demand for clean air. J Econom Manag, 5: 81–102 (1978)
Belsley D A, Kuh E, Welsch R E. Regression Diagnostics: Identifying Influential Data and Sources of Collinearity. New York: Wiley, 1980
Opsomer J D, Ruppert D. A fully automated bandwidth selection method for fitting additive models. J Amer Statist Assoc, 93: 605–619 (1998)
Fan J, Jiang J. Nonparametric inference for additive models. J Amer Statist Assoc, 100: 890–907 (2005)
Opsomer J D. Optimal bandwidth selection for fitting an additive model by local polynomial regression. PhD thesis. Ithaca, NY: Cornell University, 1995
Zhou S, Shen X, Wolfe D A. Local asymptotics for regression splines and confidence regions. Ann Statist, 26: 1760–1782 (1998)
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This work was supported by the National Natural Science Foundation of China (Grant No. 10471006)
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Jiang, J., Li, J. Two-stage local M-estimation of additive models. Sci. China Ser. A-Math. 51, 1315–1338 (2008). https://doi.org/10.1007/s11425-007-0173-6
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DOI: https://doi.org/10.1007/s11425-007-0173-6