×

zbMATH — the first resource for mathematics

Optimal estimation in additive regression models. (English) Zbl 1098.62043
Summary: This paper is concerned with optimal estimation of the additive components of a nonparametric, additive regression model. Several different smoothing methods are considered, including kernels, local polynomials, smoothing splines and orthogonal series. It is shown that, asymptotically up to first order, each additive component can be estimated as well as it could be if the other components were known. This result is used to show that in additive models the asymptotically optimal minimax rates and constants are the same as they are in nonparametric regression models with one component.

MSC:
62G08 Nonparametric regression and quantile regression
62G20 Asymptotic properties of nonparametric inference
62H12 Estimation in multivariate analysis
62C20 Minimax procedures in statistical decision theory
PDF BibTeX XML Cite
Full Text: DOI Euclid
References:
[1] Belitser, E.N. and Levit, B.Y. (1995) On minimax filtering over ellipsoids. Math. Methods Statist., 4, 259-273. · Zbl 0836.62070
[2] Efromovich, S. (1996) On nonparametric regression for iid observations in a general setting. Ann. Statist., 24, 1126-1144. · Zbl 0865.62025 · doi:10.1214/aos/1032526960
[3] Efromovich, S. and Pinsker, M.S. (1984) Learning algorithm for nonparametric filtering. Avtomat. i Telemekh., 11, 58-65.
[4] Golubev, G.K. (1987) Adaptive asymptotically minimax estimators of smooth signals. Problems Inform. Transmission, 23, 57-67. · Zbl 0636.94005
[5] Golubev, G.K. (1992) Asymptotic minimax estimation of regression in the additive model. Problemy Peredachi Informatsii, 28, 3-15 (in Russian). English translation: Problems Inform. Transmission, 28, 101-112 (1992). · Zbl 0789.62011
[6] Golubev, G.K. and Nussbaum, M. (1990) A risk bound in Sobolev class regression. Ann. Statist., 18, 758-778. · Zbl 0713.62047 · doi:10.1214/aos/1176347624
[7] Hastie, T. and Tibshirani, R. (1991) Generalized Additive Models. London: Chapman & Hall. · Zbl 0747.62061
[8] Horowitz, J. and Mammen, E. (2004) Nonparametric estimation of an additive model with a link function. Ann. Statist., 32, 2412-2443. · Zbl 1069.62035 · doi:10.1214/009053604000000814
[9] Linton, O. (2000) Efficient estimation of generalized additive nonparametric regression models. Econometric Theory, 16, 502-523. JSTOR: · Zbl 0963.62037 · doi:10.1017/S0266466600164023 · links.jstor.org
[10] Mammen, E., Linton, O. and Nielsen, J. (1999) The existence and asymptotic properties of a backfitting projection algorithm under weak conditions. Ann. Statist., 27, 1443-1490. · Zbl 0986.62028
[11] Nussbaum, M. (1985) Spline smoothing in regression models and asymptotic efficiency in L2. Ann. Statist., 13, 984-997. · Zbl 0596.62052 · doi:10.1214/aos/1176349651
[12] Opsomer, J.D. (2000) Asymptotic properties of backfitting estimators. J. Multivariate Anal., 73, 166- 179. · Zbl 1065.62506 · doi:10.1006/jmva.1999.1868
[13] Opsomer, J.D. and Ruppert, D. (1997) Fitting a bivariate additive model by local polynomial regression. Ann. Statist., 25, 185-211. · Zbl 0869.62026 · doi:10.1214/aos/1034276626
[14] Pinsker, M.S. (1980) Optimal filtering of square integrable signals in Gaussian white noise. Problemy Peredachi Informatsii, 16, 52-68 (in Russian). English translation: Problems Inform. Transmission, 16, 120-133 (1980). · Zbl 0452.94003
[15] Shorack, G.R. and Wellner, J.A. (1986) Empirical Processes with Applications to Statistics. New York: Wiley. · Zbl 1170.62365
[16] Stone, C.J. (1985) Additive regression and other nonparametric models. Ann. Statist., 13, 689-705. · Zbl 0605.62065 · doi:10.1214/aos/1176349548
[17] van de Geer, S. (2000) Empirical Processes in M-Estimation. Cambridge: Cambridge University Press. · Zbl 0953.62049
[18] Zhang, S. and Wong, M.-Y. (2003) Wavelet threshold estimation for additive regression models. Ann. Statist., 31, 152-173. · Zbl 1018.62031 · doi:10.1214/aos/1046294460
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.