Abstract
Additive regression models have a long history in multivariate non-parametric regression. They provide a model in which the regression function is decomposed as a sum of functions, each of them depending only on a single explanatory variable. The advantage of additive models over general non-parametric regression models is that they allow to obtain estimators converging at the optimal univariate rate avoiding the so-called curse of dimensionality. Beyond backfitting, marginal integration is a common procedure to estimate each component in additive models. In this paper, we propose a robust estimator of the additive components which combines local polynomials on the component to be estimated with the marginal integration procedure. The proposed estimators are consistent and asymptotically normally distributed. A simulation study allows to show the advantage of the proposal over the classical one when outliers are present in the responses, leading to estimators with good robustness and efficiency properties.
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References
Alimadad A, Salibián-Barrera M (2012) An outlier-robust fit for generalized additive models with applications to disease outbreak detection. J Am Stat Assoc 106:719–731
Baek J, Wehrly T (1993) Kernel estimation for additive models under dependence. Stoch Process Appl 47:95–112
Bianco A, Boente G (1998) Robust kernel estimators for additive models with dependent observations. Can J Stat 6:239–255
Bianco A, Boente G (2007) Robust estimators under a semiparametric partly linear autoregression model: asymptotic behavior and bandwidth selection. J Time Ser Anal 28:274–306
Boente G, Fraiman R (1989) Robust nonparametric regression estimation. J Multivar Anal 29:180–198
Boente G, Martínez A (2016) Estimating additive models with missing responses. Commun Stat Theory Methods 45:413–426
Boente G, Fraiman R, Meloche J (1997) Robust plug-in bandwidth estimators in nonparametric regression. J Stat Plan Inference 57:109–142
Boente G, González-Manteiga W, Pérez-González A (2009) Robust nonparametric estimation with missing data. J Stat Plan Inference 139:571–592
Boente G, Ruiz M, Zamar R (2010) On a robust local estimator for the scale function in heteroscedastic nonparametric regression. Stat Probab Lett 80:1185–1195
Buja A, Hastie T, Tibshirani R (1989) Linear smoothers and additive models (with discussion). Ann Stat 17:453–555
Cantoni E, Ronchetti E (2001) Resistant selection of the smoothing parameter for smoothing splines. Stat Comput 11:141–146
Chen R, Härdle W, Linton O, Serverance-Lossin E (1996) Nonparametric estimation of additive separable regression models. In: Härdle W, Schimek MG (eds) Statistical theory and computational aspects of smoothing. Proceedings of the COMPSTAT 94 satellite meeting. Springer, pp 247–265
Croux C, Gijbels I, Prosdocimi I (2011) Robust estimation of mean and dispersion functions in extended generalized additive models. Biometrics 68:31–44
Hastie TJ, Tibshirani RJ (1990) Generalized additive models. Monographs on statistics and applied probability No. 43. Chapman and Hall, London
Hengartner N, Sperlich S (2005) Rate optimal estimation with the integration method in the presence of many covariates. J Multivar Anal 95:246–272
Kong E, Linton O, Xia Y (2010) Uniform Bahadur representation for local polynomial estimates of \(M\)-regression and its application to the additive model. Econom Theory 26:1529–1564
Leung D (2005) Cross-validation in nonparametric regression with outliers. Ann Stat 33:2291–2310
Leung D, Marriott F, Wu E (1993) Bandwidth selection in robust smoothing. J Nonparametric Stat 4:333–339
Li J, Zheng Z, Zheng M (2012) Robust estimation of additive models based on marginal integration. http://www.math.pku.edu.cn:8000/var/preprint/7065
Linton O, Nielsen J (1995) A kernel method of estimating structured nonparametric regression based on marginal integration. Biometrika 82:93–101
Maronna R, Martin RD, Yohai V (2006) Robust statistics: theory and methods. Wiley, New York
Martínez-Miranda MD, Raya-Miranda R, González-Manteiga W, González-Carmona A (2008) A bootstrap local bandwidth selector for additive models. J Comput Graph Stat 17:38–55
Nielsen J, Linton O (1998) An optimization interpretation of integration and back-fitting estimators for separable nonparametric models. J R Stat Soc 60:217–222
Raya-Miranda R, Martínez-Miranda MD (2011) Data-driven local bandwidth selection for additive models with missing data. Appl Math Comput 217:10328–10342
Severance-Lossin E, Sperlich S (1999) Estimation of derivatives for additive separable models. Statistics 33:241–265
Sperlich S, Linton O, Härdle W (1999) Integration and backfitting methods in additive models-finite sample properties and comparison. TEST 8:419–458
Stone CJ (1980) Optimal rates of convergence for nonparametric estimators. Ann Stat 8:1348–1360
Stone CJ (1982) Optimal global rates of convergence for nonparametric regression. Ann Stat 10:1040–1053
Stone CJ (1985) Additive regression and other nonparametric models. Ann Stat 13:689–705
Tjøstheim D, Auestad B (1994) Nonparametric identification of nonlinear time series: selecting significant lags. J Am Stat Assoc 89:1410–1430
Wang F, Scott D (1994) The \(L_1\) method for robust nonparametric regression. J Am Stat Assoc 89:65–76
Wong RKW, Yao F, Lee TCM (2014) Robust estimation for generalized additive models. J Comput Graph Stat 23:270–289
Acknowledgments
The authors wish to thank the Associate Editor and two anonymous referees for valuable comments which led to an improved version of the original paper. This research was partially supported by Grants pip 112-201101-00339 from the Consejo Nacional de Investigaciones Científicas y Técnicas, pict 2014-0351 from the Agencia Nacional de Promoción Científica y Tecnológica and 20120130100279BA from the Universidad de Buenos Aires at Buenos Aires, Argentina.
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Boente, G., Martínez, A. Marginal integration M-estimators for additive models. TEST 26, 231–260 (2017). https://doi.org/10.1007/s11749-016-0508-0
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DOI: https://doi.org/10.1007/s11749-016-0508-0