Wavelet threshold estimation for additive regression models
Man-Yu Wong and Shuanglin Zhang
Source: Ann. Statist. Volume 31, Number 1
(2003), 152-173.
Abstract
Additive regression models have turned out to be useful statistical tools in the analysis of high-dimensional data. The attraction of such models is that the additive component can be estimated with the same optimal convergence rate as a one-dimensional nonparametric regression. However, this optimal property holds only when all the additive components have the same degree of "homogeneous" smoothness. In this paper, we propose a two-step wavelet thresholding estimation process in which the estimator is adaptive to different degrees of smoothness in different components and also adaptive to the "inhomogeneous" smoothness described by the Besov space. The estimator of an additive component constructed by the proposed procedure is shown to attain the one-dimensional optimal convergence rate even when the components have different degrees of "inhomogeneous" smoothness.
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Keywords: Local polynomial estimation; wavelet estimation; optimal convergence rate; additive regression; threshold; Besov space
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Permanent link to this document: http://projecteuclid.org/euclid.aos/1046294460
Digital Object Identifier: doi:10.1214/aos/1046294460
Mathematical Reviews number (MathSciNet): MR1962502
Zentralblatt MATH identifier: 1018.62031
References
COHEN, A., DAUBECHIES, I. and VIAL, P. (1993). Wavelets on the interval and fast wavelet transforms. Appl. Comput. Harmon. Anal. 1 54-81.
Zentralblatt MATH: 0795.42018
Mathematical Reviews (MathSciNet): MR1256527
Digital Object Identifier: doi:10.1006/acha.1993.1005
DAUBECHIES, I. (1992). Ten Lectures on Wavelets. SIAM, Philadelphia.
Mathematical Reviews (MathSciNet): MR93e:42045
DELy ON, B. and JUDITSKY, A. (1996). On minimax wavelet estimators. Appl. Comput. Harmon. Anal. 3 215-228.
Mathematical Reviews (MathSciNet): MR98k:62051
Zentralblatt MATH: 0865.62023
Digital Object Identifier: doi:10.1006/acha.1996.0017
DONOHO, D. L. (1995). De-noising by soft-thresholding. IEEE Trans. Inform. Theory 41 613-627.
Mathematical Reviews (MathSciNet): MR96b:94002
Digital Object Identifier: doi:10.1109/18.382009
DONOHO, D. L. and JOHNSTONE, I. M. (1994). Ideal spatial adaptation by wavelet shrinkage. Biometrika 81 425-455.
Mathematical Reviews (MathSciNet): MR95m:62076
Zentralblatt MATH: 0815.62019
Digital Object Identifier: doi:10.1093/biomet/81.3.425
JSTOR: links.jstor.org
DONOHO, D. L. and JOHNSTONE, I. M. (1995). Adapting to unknown smoothness via wavelet shrinkage. J. Amer. Statist. Assoc. 90 1200-1224.
Mathematical Reviews (MathSciNet): MR96k:62093
Zentralblatt MATH: 0869.62024
Digital Object Identifier: doi:10.2307/2291512
JSTOR: links.jstor.org
DONOHO, D. L. and JOHNSTONE, I. M. (1998). Minimax estimation via wavelet shrinkage. Ann. Statist. 26 879-921.
Mathematical Reviews (MathSciNet): MR99i:62086
Zentralblatt MATH: 0935.62041
Digital Object Identifier: doi:10.1214/aos/1024691081
Project Euclid: euclid.aos/1024691081
DONOHO, D. L., JOHNSTONE, I. M., KERKy ACHARIAN, G. and PICARD, D. (1995). Wavelet shrinkage: Asy mptopia? J. Roy. Statist. Soc. Ser. B 57 301-369.
Mathematical Reviews (MathSciNet): MR96g:62068
JSTOR: links.jstor.org
DONOHO, D. L., JOHNSTONE, I. M., KERKy ACHARIAN, G. and PICARD, D. (1996). Density estimation by wavelet thresholding. Ann. Statist. 24 508-539.
Mathematical Reviews (MathSciNet): MR97f:62061
Zentralblatt MATH: 0860.62032
Digital Object Identifier: doi:10.1214/aos/1032894451
Project Euclid: euclid.aos/1032894451
FAN, J., HÄRDLE, W. and MAMMEN, E. (1998). Direct estimation of low-dimensional components in additive models. Ann. Statist. 26 943-971.
Zentralblatt MATH: 01359546
FAN, J. and ZHANG, W. (1999). Statistical estimation in varying coefficient models. Ann. Statist. 27 1491-1518.
Zentralblatt MATH: 0977.62039
Mathematical Reviews (MathSciNet): MR1742497
Digital Object Identifier: doi:10.1214/aos/1017939139
Project Euclid: euclid.aos/1017939139
FRIEDMAN, J. H. and STUETZLE, W. (1981). Project pursuit regression. J. Amer. Statist. Assoc. 76 817-823.
Mathematical Reviews (MathSciNet): MR650892
Digital Object Identifier: doi:10.2307/2287576
JSTOR: links.jstor.org
HALL, P., KERKy ACHARIAN, G. and PICARD, D. (1998). Block threshold rules for curve estimation using kernel and wavelet methods. Ann. Statist. 26 922-942.
Zentralblatt MATH: 0929.62040
Mathematical Reviews (MathSciNet): MR1635418
Digital Object Identifier: doi:10.1214/aos/1024691082
Project Euclid: euclid.aos/1024691082
HALL, P. and TURLACH, B. A. (1997). Interpolation methods for nonlinear wavelet regression with irregularly spaced design. Ann. Statist. 25 1912-1925.
Zentralblatt MATH: 0881.62044
Mathematical Reviews (MathSciNet): MR1474074
Digital Object Identifier: doi:10.1214/aos/1069362378
Project Euclid: euclid.aos/1069362378
HASTIE, T. J. and TIBSHIRANI, R. J. (1990). Generalized Additive Models. Chapman and Hall, London.
Mathematical Reviews (MathSciNet): MR92e:62117
LINTON, O. B. (1996). Estimation of additive regression models with known links. Biometrika 83 529-540.
Mathematical Reviews (MathSciNet): MR98c:62068
Zentralblatt MATH: 0866.62017
Digital Object Identifier: doi:10.1093/biomet/83.3.529
JSTOR: links.jstor.org
LINTON, O. B. (1997). Efficient estimation of additive nonparametric regression models. Biometrika 84 469-473.
Mathematical Reviews (MathSciNet): MR98k:62062
Zentralblatt MATH: 0882.62038
Digital Object Identifier: doi:10.1093/biomet/84.2.469
JSTOR: links.jstor.org
LINTON, O. B. and NIELSEN, J. P. (1995). A kernel method of estimating structured nonparametric regression based on marginal integration. Biometrika 82 93-100.
Mathematical Reviews (MathSciNet): MR96f:62065
Zentralblatt MATH: 0823.62036
Digital Object Identifier: doi:10.1093/biomet/82.1.93
JSTOR: links.jstor.org
NEUMANN, M. H. and VON SACHS, R. (1995). Wavelet thresholding: Bey ond the Gaussian i.i.d. situation. Wavelets and Statistics. Lecture Notes in Statist. 103 301-329. Springer, Berlin.
NIELSEN, J. P. and LINTON, O. B. (1998). An optimization interpretation of integration and back-fitting estimators for separable nonparametric models. J. R. Stat. Soc. Ser. B Stat. Methodol. 60 217-222.
Mathematical Reviews (MathSciNet): MR99a:62054
Zentralblatt MATH: 0909.62040
Digital Object Identifier: doi:10.1111/1467-9868.00120
JSTOR: links.jstor.org
OPSOMER, J. and RUPPERT, D. (1997). Fitting a bivariate additive model by local poly nomial regression. Ann. Statist. 25 186-211.
Mathematical Reviews (MathSciNet): MR98b:62074
Zentralblatt MATH: 0869.62026
Digital Object Identifier: doi:10.1214/aos/1034276626
Project Euclid: euclid.aos/1034276626
PETROV, V. V. (1975). Sums of Independent Random Variables. Springer, New York.
Mathematical Reviews (MathSciNet): MR388499
POLLARD, D. (1984). Convergence of Stochastic Processes. Springer, New York.
Mathematical Reviews (MathSciNet): MR86i:60074
Zentralblatt MATH: 0544.60045
STONE, C. J. (1985). Additive regression and other nonparametric models. Ann. Statist. 13 689-705.
Zentralblatt MATH: 0605.62065
Mathematical Reviews (MathSciNet): MR790566
Digital Object Identifier: doi:10.1214/aos/1176349548
Project Euclid: euclid.aos/1176349548
STONE, C. J. (1986). The dimensionality reduction principle for generalized additive models. Ann. Statist. 14 590-606.
Zentralblatt MATH: 0603.62050
Mathematical Reviews (MathSciNet): MR840516
Digital Object Identifier: doi:10.1214/aos/1176349940
Project Euclid: euclid.aos/1176349940
STONE, C. J. (1994). The use of poly nomial splines and their tensor products in multivariate function estimation (with discussion). Ann. Statist. 22 118-184.
Mathematical Reviews (MathSciNet): MR1272079
Zentralblatt MATH: 0827.62038
Digital Object Identifier: doi:10.1214/aos/1176325361
Project Euclid: euclid.aos/1176325361
TJØSTHEIM, D. and AUESTAD, B. H. (1994). Nonparametric identification of nonlinear time series: Projections. J. Amer. Statist. Assoc. 89 1398-1409.
TRIEBEL, H. (1992). Theory of Function Spaces II. Birkhäuser, Boston.
Mathematical Reviews (MathSciNet): MR93f:46029
Zentralblatt MATH: 0763.46025
ZHANG, S., WONG, M.-Y. and ZHENG, Z. (2002). Wavelet threshold estimation of a regression function with random design. J. Multivariate Anal. 80 256-284.
Zentralblatt MATH: 1003.62040
Mathematical Reviews (MathSciNet): MR1889776
Digital Object Identifier: doi:10.1006/jmva.2000.1980
HOUGHTON, MICHIGAN 49931 AND DEPARTMENT OF MATHEMATICS HEILONGJIANG UNIVERSITY HARBIN 150080 CHINA DEPARTMENT OF MATHEMATICS HONG KONG UNIVERSITY OF SCIENCE AND TECHNOLOGY
CLEAR WATER BAY, KOWLOON HONG KONG E-MAIL: mamy wong@ust.hk
