×

zbMATH — the first resource for mathematics

An adaptive two-stage estimation method for additive models. (English) Zbl 1194.62048
An additive nonparametric regression model is considered of the form \[ Y=\mu+r_1(x^1)+\dots+r_d(x^d)+\varepsilon, \] where \(r_i\) are unknown functions to be estimated. A two-stage estimation procedure is proposed. At the first stage, projective estimates \(\tilde r_i\) are derived. At the second stage multiplicative (\(\tilde r_i(x)\xi_i(x)\)) or additive (\(\tilde r_i(x)+\zeta_i(x)\)) improvements are made which minimize the (estimated) locally smoothed \(L_2\) distances from the true \(r_i\). It is shown that optimal rates of MSE convergence can be derived for these estimates, and for \(r_i\) for some “good” functional spaces the estimates can achieve the “parametric” rate MSE\(\sim O(n^{-1})\). A cross-validation algorithm is proposed for optimal bandwidth selection. Results of simulations are presented.

MSC:
62G08 Nonparametric regression and quantile regression
62G20 Asymptotic properties of nonparametric inference
65C60 Computational problems in statistics (MSC2010)
Software:
gamair
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Barry, Testing for additivity of a regression function, Ann. Statist. 21 pp 235– (1993) · Zbl 0771.62033
[2] Buja, Linear smoothers and additive models (with discussion), Ann. Statist. 17 pp 453– (1989)
[3] Chen, Statistical theory and computational aspects of smoothing pp 247– (1996) · doi:10.1007/978-3-642-48425-4_18
[4] Eubank, Testing for additivity in non-parametric regression, Ann. Statist. 23 pp 1896– (1995) · Zbl 0858.62036 · doi:10.1214/aos/1034713639
[5] Fan, Direct estimation of low-dimensional components in additive models, Ann. Statist. 26 pp 943– (1998) · Zbl 1073.62527
[6] Friedman, Multidimensional additive spline approximation, SIAM J. Sci. Statist. Comput. 4 pp 291– (1983) · Zbl 0525.65006
[7] Friedman, Projection pursuit regression, J. Amer. Statist. Assoc. 76 pp 817– (1981)
[8] Glad, Parametrically guided non-parametric regression, Scand. J. Statist. 25 pp 649– (1998) · Zbl 0927.62037
[9] Härdle, Optimal bandwidth selection in nonparametric regression function estimation, Ann. Statist. 13 pp 1465– (1985) · Zbl 0594.62043
[10] Hart, Nonparametric smoothing and lack-of-fit tests (1997) · Zbl 0886.62043 · doi:10.1007/978-1-4757-2722-7
[11] Hastie, Generalized additive models (1990) · Zbl 0747.62061
[12] Hjort, Nonparametric density estimation with a parametric start, Ann. Statist. 23 pp 882– (1995) · Zbl 0838.62027
[13] Hjort, Locally parametric nonparametric density estimation, Ann Statist. 24 pp 1619– (1996) · Zbl 0867.62030
[14] Horowitz, Nonparametric estimation of an additive model with a link function, Ann. Statist. 32 pp 2412– (2004) · Zbl 1069.62035
[15] Linton, Estimation of additive regression models with known links, Biometrika 83 pp 529– (1996) · Zbl 0866.62017
[16] Linton, A kernel method of estimating structured nonparametric regression based on marginal integration, Biometrika 82 pp 93– (1995) · Zbl 0823.62036
[17] Naito, Semiparametric density estimation by local L2-fitting, Ann. Statist. 32 pp 1162– (2004) · Zbl 1091.62023
[18] Stone, Additive regression and other nonparametric models, Ann. Statist. 13 pp 685– (1985) · Zbl 0605.62065
[19] Tong, Estimating residual variance in nonparametric regression using least squares, Biometrika 92 pp 821– (2005) · Zbl 1151.62318
[20] Wood, Generalized additive models: an introduction (2006) · Zbl 1087.62082
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.