Source: Ann. Statist.
Volume 31, Number 1
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.
COHEN, A., DAUBECHIES, I. and VIAL, P. (1993). Wavelets on the interval and fast wavelet transforms. Appl. Comput. Harmon. Anal. 1 54-81.
DAUBECHIES, I. (1992). Ten Lectures on Wavelets. SIAM, Philadelphia.
DELy ON, B. and JUDITSKY, A. (1996). On minimax wavelet estimators. Appl. Comput. Harmon. Anal. 3 215-228.
DONOHO, D. L. (1995). De-noising by soft-thresholding. IEEE Trans. Inform. Theory 41 613-627.
DONOHO, D. L. and JOHNSTONE, I. M. (1994). Ideal spatial adaptation by wavelet shrinkage. Biometrika 81 425-455.
DONOHO, D. L. and JOHNSTONE, I. M. (1995). Adapting to unknown smoothness via wavelet shrinkage. J. Amer. Statist. Assoc. 90 1200-1224.
DONOHO, D. L. and JOHNSTONE, I. M. (1998). Minimax estimation via wavelet shrinkage. Ann. Statist. 26 879-921.
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.
DONOHO, D. L., JOHNSTONE, I. M., KERKy ACHARIAN, G. and PICARD, D. (1996). Density estimation by wavelet thresholding. Ann. Statist. 24 508-539.
FAN, J., HÄRDLE, W. and MAMMEN, E. (1998). Direct estimation of low-dimensional components in additive models. Ann. Statist. 26 943-971.
FAN, J. and ZHANG, W. (1999). Statistical estimation in varying coefficient models. Ann. Statist. 27 1491-1518.
FRIEDMAN, J. H. and STUETZLE, W. (1981). Project pursuit regression. J. Amer. Statist. Assoc. 76 817-823.
Mathematical Reviews (MathSciNet): MR650892
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.
HALL, P. and TURLACH, B. A. (1997). Interpolation methods for nonlinear wavelet regression with irregularly spaced design. Ann. Statist. 25 1912-1925.
HASTIE, T. J. and TIBSHIRANI, R. J. (1990). Generalized Additive Models. Chapman and Hall, London.
LINTON, O. B. (1996). Estimation of additive regression models with known links. Biometrika 83 529-540.
LINTON, O. B. (1997). Efficient estimation of additive nonparametric regression models. Biometrika 84 469-473.
LINTON, O. B. and NIELSEN, J. P. (1995). A kernel method of estimating structured nonparametric regression based on marginal integration. Biometrika 82 93-100.
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.
OPSOMER, J. and RUPPERT, D. (1997). Fitting a bivariate additive model by local poly nomial regression. Ann. Statist. 25 186-211.
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.
STONE, C. J. (1985). Additive regression and other nonparametric models. Ann. Statist. 13 689-705.
Mathematical Reviews (MathSciNet): MR790566
STONE, C. J. (1986). The dimensionality reduction principle for generalized additive models. Ann. Statist. 14 590-606.
Mathematical Reviews (MathSciNet): MR840516
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.
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.
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.
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 email@example.com