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Nonparametric estimation of an additive model with a link function. (English) Zbl 1069.62035
Summary: This paper describes an estimator of the additive components of a nonparametric additive model with a known link function. When the additive components are twice continuously differentiable, the estimator is asymptotically normally distributed with a rate of convergence in probability of \(n^{-2/5}\). This is true regardless of the (finite) dimension of the explanatory variable. Thus, in contrast to the existing asymptotically normal estimator, the new estimator has no curse of dimensionality. Moreover, the estimator has an oracle property. The asymptotic distribution of each additive component is the same as it would be if the other components were known with certainty.

MSC:
62G08 Nonparametric regression and quantile regression
62G20 Asymptotic properties of nonparametric inference
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[1] Bosq, D. (1998). Nonparametric Statistics for Stochastic Procesess. Estimation and Prediction , 2nd ed. Lecture Notes in Statist. 110 . Springer, Berlin. · Zbl 0902.62099 · doi:10.1007/978-1-4612-1718-3
[2] Breiman, L. and Friedman, J. H. (1985). Estimating optimal transformations for multiple regression and correlation (with discussion). J. Amer. Statist. Assoc. 80 580–619. · Zbl 0594.62044 · doi:10.2307/2288473
[3] Buja, A., Hastie, T. and Tibshirani, R. J. (1989). Linear smoothers and additive models (with discussion). Ann. Statist. 17 453–555. JSTOR: · Zbl 0689.62029 · doi:10.1214/aos/1176347115 · links.jstor.org
[4] Chen, R., Härdle, W., Linton, O. B. and Severance-Lossin, E. (1996). Nonparametric estimation of additive separable regression models. In Statistical Theory and Computational Aspects of Smoothing (W. Härdle and M. Schimek, eds.) 247–253. Physica, Heidelberg.
[5] Fan, J. and Chen, J. (1999). One-step local quasi-likelihood estimation. J. R. Stat. Soc. Ser. B Stat. Methodol. 61 927–943. · Zbl 0940.62039 · doi:10.1111/1467-9868.00211
[6] Fan, J. and Gijbels, I. (1996). Local Polynomial Modelling and Its Applications . Chapman and Hall, London. · Zbl 0873.62037
[7] Fan, J., Härdle, W. and Mammen, E. (1998). Direct estimation of low-dimensional components in additive models. Ann. Statist. 26 943–971. · Zbl 1073.62527 · doi:10.1214/aos/1024691083
[8] Hastie, T. J. and Tibshirani, R. J. (1990). Generalized Additive Models . Chapman and Hall, London. · Zbl 0747.62061
[9] Horowitz, J. L., Klemelä, J. and Mammen, E. (2002). Optimal estimation in additive regression models. Working paper, Institut für Angewandte Mathematik, Ruprecht-Karls-Univertsität, Heidelberg, Germany.
[10] Linton, O. B. and Härdle, W. (1996). Estimating additive regression models with known links. Biometrika 83 529–540. · Zbl 0866.62017 · doi:10.1093/biomet/83.3.529 · www3.oup.co.uk
[11] Linton, O. B. and Nielsen, J. P. (1995). A kernel method of estimating structured nonparametric regression based on marginal integration. Biometrika 82 93–100. · Zbl 0823.62036 · doi:10.1093/biomet/82.1.93
[12] Mammen, E., Linton, O. B. and Nielsen, J. P. (1999). The existence and asymptotic properties of backfitting projection algorithm under weak conditions. Ann. Statist. 27 1443–1490. · Zbl 0986.62028
[13] Newey, W. K. (1997). Convergence rates and asymptotic normality for series estimators. J. Econometrics 79 147–168. · Zbl 0873.62049 · doi:10.1016/S0304-4076(97)00011-0
[14] Opsomer, J. D. (2000). Asymptotic properties of backfitting estimators. J. Multivariate Anal. 73 166–179. · Zbl 1065.62506 · doi:10.1006/jmva.1999.1868
[15] Opsomer, J. D. and Ruppert, D. (1997). Fitting a bivariate additive model by local polynomial regression. Ann. Statist. 25 186–211. · Zbl 0869.62026 · doi:10.1214/aos/1034276626
[16] Pollard, D. (1984). Convergence of Stochastic Processes. Springer, New York. · Zbl 0544.60045
[17] Serfling, R. J. (1980). Approximation Theorems of Mathematical Statistics . Wiley, New York. · Zbl 0538.62002
[18] Stone, C. J. (1985). Additive regression and other nonparametric models. Ann. Statist. 13 689–705. JSTOR: · Zbl 0605.62065 · doi:10.1214/aos/1176349548 · links.jstor.org
[19] Stone, C. J. (1986). The dimensionality reduction principle for generalized additive models. Ann. Statist. 14 590–606. JSTOR: · Zbl 0603.62050 · doi:10.1214/aos/1176349940 · links.jstor.org
[20] Stone, C. J. (1994). The use of polynomial splines and their tensor products in multivariate function estimation (with discussion). Ann. Statist. 22 118–184. JSTOR: · Zbl 0827.62038 · doi:10.1214/aos/1176325361 · links.jstor.org
[21] Tjøstheim, D. and Auestad, B. H. (1994). Nonparametric identification of nonlinear time series: Projections. J. Amer. Statist. Assoc. 89 1398–1409. · Zbl 0813.62036 · doi:10.2307/2291002
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