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Zhang, Shuanglin; Wong, Man-Yu

Wavelet threshold estimation for additive regression models. (English) Zbl 1018.62031

Ann. Stat. 31, No. 1, 152-173 (2003).
Summary: 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.
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.

Cited in 8 Documents

MSC:

62G08 Nonparametric regression and quantile regression
62G20 Asymptotic properties of nonparametric inference

Keywords:

local polynomial estimation; wavelet estimation; optimal convergence rate; additive regression; threshold; Besov space
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References:

[1] COHEN, A., DAUBECHIES, I. and VIAL, P. (1993). Wavelets on the interval and fast wavelet transforms. Appl. Comput. Harmon. Anal. 1 54-81. · Zbl 0795.42018 · doi:10.1006/acha.1993.1005
[2] DAUBECHIES, I. (1992). Ten Lectures on Wavelets. SIAM, Philadelphia. · Zbl 0776.42018
[3] DELy ON, B. and JUDITSKY, A. (1996). On minimax wavelet estimators. Appl. Comput. Harmon. Anal. 3 215-228. · Zbl 0865.62023 · doi:10.1006/acha.1996.0017
[4] DONOHO, D. L. (1995). De-noising by soft-thresholding. IEEE Trans. Inform. Theory 41 613-627. · Zbl 0820.62002 · doi:10.1109/18.382009
[5] DONOHO, D. L. and JOHNSTONE, I. M. (1994). Ideal spatial adaptation by wavelet shrinkage. Biometrika 81 425-455. JSTOR: · Zbl 0815.62019 · doi:10.1093/biomet/81.3.425
[6] DONOHO, D. L. and JOHNSTONE, I. M. (1995). Adapting to unknown smoothness via wavelet shrinkage. J. Amer. Statist. Assoc. 90 1200-1224. JSTOR: · Zbl 0869.62024 · doi:10.2307/2291512
[7] DONOHO, D. L. and JOHNSTONE, I. M. (1998). Minimax estimation via wavelet shrinkage. Ann. Statist. 26 879-921. · Zbl 0935.62041 · doi:10.1214/aos/1024691081
[8] 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. JSTOR:
[9] DONOHO, D. L., JOHNSTONE, I. M., KERKy ACHARIAN, G. and PICARD, D. (1996). Density estimation by wavelet thresholding. Ann. Statist. 24 508-539. · Zbl 0860.62032 · doi:10.1214/aos/1032894451
[10] 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
[11] FAN, J. and ZHANG, W. (1999). Statistical estimation in varying coefficient models. Ann. Statist. 27 1491-1518. · Zbl 0977.62039 · doi:10.1214/aos/1017939139
[12] FRIEDMAN, J. H. and STUETZLE, W. (1981). Project pursuit regression. J. Amer. Statist. Assoc. 76 817-823. JSTOR: · doi:10.2307/2287576
[13] 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. · Zbl 0929.62040 · doi:10.1214/aos/1024691082
[14] HALL, P. and TURLACH, B. A. (1997). Interpolation methods for nonlinear wavelet regression with irregularly spaced design. Ann. Statist. 25 1912-1925. · Zbl 0881.62044 · doi:10.1214/aos/1069362378
[15] HASTIE, T. J. and TIBSHIRANI, R. J. (1990). Generalized Additive Models. Chapman and Hall, London. · Zbl 0747.62061
[16] LINTON, O. B. (1996). Estimation of additive regression models with known links. Biometrika 83 529-540. JSTOR: · Zbl 0866.62017 · doi:10.1093/biomet/83.3.529
[17] LINTON, O. B. (1997). Efficient estimation of additive nonparametric regression models. Biometrika 84 469-473. JSTOR: · Zbl 0882.62038 · doi:10.1093/biomet/84.2.469
[18] LINTON, O. B. and NIELSEN, J. P. (1995). A kernel method of estimating structured nonparametric regression based on marginal integration. Biometrika 82 93-100. JSTOR: · Zbl 0823.62036 · doi:10.1093/biomet/82.1.93
[19] 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. · Zbl 0831.62071
[20] 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. JSTOR: · Zbl 0909.62040 · doi:10.1111/1467-9868.00120
[21] OPSOMER, J. and RUPPERT, D. (1997). Fitting a bivariate additive model by local poly nomial regression. Ann. Statist. 25 186-211. · Zbl 0869.62026 · doi:10.1214/aos/1034276626
[22] PETROV, V. V. (1975). Sums of Independent Random Variables. Springer, New York. · Zbl 0322.60043
[23] POLLARD, D. (1984). Convergence of Stochastic Processes. Springer, New York. · Zbl 0544.60045
[24] STONE, C. J. (1985). Additive regression and other nonparametric models. Ann. Statist. 13 689-705. · Zbl 0605.62065 · doi:10.1214/aos/1176349548
[25] STONE, C. J. (1986). The dimensionality reduction principle for generalized additive models. Ann. Statist. 14 590-606. · Zbl 0603.62050 · doi:10.1214/aos/1176349940
[26] 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. · Zbl 0827.62038 · doi:10.1214/aos/1176325361
[27] 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
[28] TRIEBEL, H. (1992). Theory of Function Spaces II. Birkhäuser, Boston. · Zbl 0763.46025
[29] 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. · Zbl 1003.62040 · doi:10.1006/jmva.2000.1980
[30] 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
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