Adaptive estimation of an additive regression function from weakly dependent data. (English) Zbl 1302.62092
Summary: A \(d\)-dimensional nonparametric additive regression model with dependent observations is considered. Using the marginal integration technique and wavelets methodology, we develop a new adaptive estimator for a component of the additive regression function. Its asymptotic properties are investigated via the minimax approach under the \(\mathbb{L}_2\) risk over Besov balls. We prove that it attains a sharp rate of convergence which turns to be the one obtained in the i.i.d. case for the standard univariate regression estimation problem.
MSC:
| 62G08 | Nonparametric regression and quantile regression |
| 62G20 | Asymptotic properties of nonparametric inference |
References:
| [1] | Amato, U.; Antoniadis, A., Adaptive wavelet series estimation in separable nonparametric regression models, Stat. Comput., 11, 373-394 (2001) |
| [2] | Amato, U.; Antoniadis, A.; De Feis, I., Fourier series approximation of separable models, J. Comput. Appl. Math., 146, 459-479 (2002) · Zbl 1058.62034 |
| [3] | Bradley, R. C., Introduction to Strong Mixing Conditions. Vols. 1,2,3 (2007), Kendrick Press · Zbl 1134.60004 |
| [4] | Buja, A.; Hastie, T.; Tibshirani, R., Linear smoothers and additive models (with discussion), Ann. Statist., 17, 453-555 (1989) · Zbl 0689.62029 |
| [5] | Camlong-Viot, C.; Rodrìguez-Pòo, J. M.; Vieu, P., Nonparametric and semiparametric estimation of additive models with both discrete and continuous variables under dependence, (The Art of Semiparametrics. The Art of Semiparametrics, Contrib. Statist. (2006), Physica-Verlag/Springer: Physica-Verlag/Springer Heidelberg), 155-178 · Zbl 1271.62064 |
| [6] | Carrasco, M.; Chen, X., Mixing and moment properties of various GARCH and stochastic volatility models, Econometric Theory, 18, 17-39 (2002) · Zbl 1181.62125 |
| [7] | Chesneau, C., On the adaptive wavelet estimation of a multidimensional regression function under \(\alpha \)-mixing dependence: beyond the standard assumptions on the noise, Comment. Math. Univ. Carolin., 4, 527-556 (2013) · Zbl 1313.62058 |
| [8] | Chesneau, C., A general result on the mean integrated squared error of the hard thresholding wavelet estimator under \(\alpha \)-mixing dependence, J. Probab. Stat., 2014 (2014), Article ID 403764, 12 pages · Zbl 1307.62103 |
| [9] | Cohen, A.; Daubechies, I.; Jawerth, B.; Vial, P., Wavelets on the interval and fast wavelet transforms, Appl. Comput. Harmon. Anal., 24, 1, 54-81 (1993) · Zbl 0795.42018 |
| [10] | Davydov, Y., The invariance principle for stationary processes, Theory Probab. Appl., 15, 3, 498-509 (1970) · Zbl 0219.60030 |
| [11] | Debbarh, M., Asymptotic normality for the wavelets estimator of the additive regression components, C. R. Math., 343, 9, 601-606 (2006), 1 · Zbl 1101.62038 |
| [12] | Debbarh, M.; Maillot, B., Additive regression model for continuous time processes, Comm. Statist. Theory Methods, 37, 13-15, 2416-2432 (2008) · Zbl 1259.62079 |
| [13] | Debbarh, M.; Maillot, B., Asymptotic normality of the additive regression components for continuous time processes, C. R. Math. Acad. Sci. Paris, 346, 15-16, 901-906 (2008) · Zbl 1144.62076 |
| [14] | Delyon, B.; Juditsky, A., On minimax wavelet estimators, Appl. Comput. Harmon. Anal., 3, 215-228 (1996) · Zbl 0865.62023 |
| [15] | Donoho, D. L.; Johnstone, I. M.; Kerkyacharian, G.; Picard, D., Density estimation by wavelet thresholding, Ann. Statist., 24, 508-539 (1996) · Zbl 0860.62032 |
| [16] | Doukhan, P., (Mixing. Properties and Examples. Mixing. Properties and Examples, Lecture Notes in Statistics, vol. 85 (1994), Springer Verlag: Springer Verlag New York) · Zbl 0801.60027 |
| [17] | Fan, J.; Jiang, J., Nonparametric inferences for additive models, J. Amer. Statist. Assoc., 100, 890-907 (2005) · Zbl 1117.62328 |
| [18] | Gao, J.; Tong, H.; Wolff, R., Adaptive orthogonal series estimation in additive stochastic regression models, Statist. Sinica, 12, 2, 409-428 (2002) · Zbl 0998.62037 |
| [19] | Härdle, W., Applied Nonparametric Regression (1990), Cambridge University Press · Zbl 0714.62030 |
| [20] | Härdle, W.; Kerkyacharian, G.; Picard, D.; Tsybakov, A., (Wavelet, Approximation and Statistical Applications. Wavelet, Approximation and Statistical Applications, Lectures Notes in Statistics, vol. 129 (1998), Springer Verlag: Springer Verlag New York) · Zbl 0899.62002 |
| [21] | Hastie, T. J.; Tibshirani, R. J., Generalized Additive Models (1990), Chapman and Hall: Chapman and Hall London · Zbl 0747.62061 |
| [22] | Liebscher, E., Estimation of the density and the regression function under mixing conditions, Statist. Decisions, 19, 1, 9-26 (2001) · Zbl 1179.62051 |
| [23] | Linton, O. B., Efficient estimation of additive nonparametric regression models, Biometrika, 84, 469-473 (1997) · Zbl 0882.62038 |
| [24] | Mallat, S., A Wavelet Tour of Signal Processing. The Sparse Way (2009), Elsevier/Academic Press: Elsevier/Academic Press Amsterdam, With contributions from Gabriel Peyré · Zbl 1170.94003 |
| [25] | Masry, E., Strong consistency and rates for deconvolution of multivariate densities of stationary processes, Stochastic Process. Appl., 47, 53-74 (1993) · Zbl 0797.62071 |
| [26] | Meyer, Y., Wavelets and Operators (1992), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0776.42019 |
| [27] | Modha, D.; Masry, E., Minimum complexity regression estimation with weakly dependent observations, IEEE Trans. Inform. Theory, 42, 2133-2145 (1996) · Zbl 0868.62015 |
| [28] | Newey, W. K., Kernel estimation of partial means and a general variance estimator, Econometric Theory, 10, 2, 233-253 (1994) |
| [29] | Opsomer, J. D.; Ruppert, D., Fitting a bivariate additive model by local polynomial regression, Ann. Statist., 25, 186-211 (1997) · Zbl 0869.62026 |
| [30] | Opsomer, J. D.; Ruppert, D., A fully automated bandwidth selection method for fitting additive models, J. Amer. Statist. Assoc., 93, 605-619 (1998) · Zbl 0953.62034 |
| [31] | Patil, P. N.; Truong, Y. K., Asymptotics for wavelet based estimates of piecewise smooth regression for stationary time series, Ann. Inst. Statist. Math., 53, 1, 159-178 (2001) · Zbl 0995.62092 |
| [32] | Sardy, S.; Tseng, P., AMlet, RAMlet, and GAMlet: automatic nonlinear fitting of additive models, robust and generalized, with wavelets, J. Comput. Graph. Statist., 13, 283-309 (2004) |
| [33] | Sperlich, S.; Tjostheim, D.; Yang, L., Nonparametric estimation and testing of interaction in additive models, Econometric Theory, 18, 197-251 (2002) · Zbl 1109.62310 |
| [34] | Stone, C. J., Additive regression and other nonparametric models, Ann. Statist., 13, 689-705 (1985) · Zbl 0605.62065 |
| [35] | Stone, C. J., The dimensionality reduction principle for generalized additive models, Ann. Statist., 14, 590-606 (1986) · Zbl 0603.62050 |
| [36] | Stone, C. J., The use of polynomial splines and their tensor products in multivariate function estimation (with discussion), Ann. Statist., 22, 118-184 (1994) · Zbl 0827.62038 |
| [37] | Tsybakov, A. B., Introduction à l’Estimation non Paramétrique (2004), Springer · Zbl 1029.62034 |
| [38] | White, H.; Domowitz, I., Nonlinear regression with dependent observations, Econometrica, 52, 143-162 (1984) · Zbl 0533.62055 |
| [39] | Withers, C. S., Conditions for linear processes to be strong-mixing, Z. Wahrscheinlichkeitstheor. Verwandte Geb., 57, 477-480 (1981) · Zbl 0465.60032 |
| [40] | Zhang, S.; Wong, M.-Y., Wavelet threshold estimation for additive regression models, Ann. Statist., 31, 152-173 (2003) · Zbl 1018.62031 |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
