The Annals of Statistics

A data-driven block thresholding approach to wavelet estimation

T. Tony Cai and Harrison H. Zhou

Full-text: Open access


A data-driven block thresholding procedure for wavelet regression is proposed and its theoretical and numerical properties are investigated. The procedure empirically chooses the block size and threshold level at each resolution level by minimizing Stein’s unbiased risk estimate. The estimator is sharp adaptive over a class of Besov bodies and achieves simultaneously within a small constant factor of the minimax risk over a wide collection of Besov Bodies including both the “dense” and “sparse” cases. The procedure is easy to implement. Numerical results show that it has superior finite sample performance in comparison to the other leading wavelet thresholding estimators.

Article information

Ann. Statist. Volume 37, Number 2 (2009), 569-595.

First available in Project Euclid: 10 March 2009

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G08: Nonparametric regression
Secondary: 62G20: Asymptotic properties

Adaptivity Besov body block thresholding James–Stein estimator nonparametric regression Stein’s unbiased risk estimate wavelets


Cai, T. Tony; Zhou, Harrison H. A data-driven block thresholding approach to wavelet estimation. Ann. Statist. 37 (2009), no. 2, 569--595. doi:10.1214/07-AOS538.

Export citation


  • Antoniadis, A. and Fan, J. (2001). Regularization of wavelet approximations (with discussion). J. Amer. Statist. Assoc. 96 939–967.
  • Breiman, L. (1995). Better subset regression using the nonnegative garrote. Technometrics 37 373–384.
  • Cai, T. (1999). Adaptive wavelet estimation: A block thresholding and oracle inequality approach. Ann. Statist. 27 898–924.
  • Cai, T., Low, M. G. and Zhao, L. (2000). Blockwise thresholding methods for sharp adaptive estimation. Technical report, Dept. Statistics, Univ. Pennsylvania.
  • Cai, T. and Silverman, B. W. (2001). Incorporating information on neighboring coefficients into wavelet estimation. Sankhyā Ser. B 63 127–148.
  • Cai, T. and Zhou, H. (2005). A data-driven block thresholding approach to wavelet estimation. Technical report, Dept. Statistics, Univ. Pennsylvania.
  • Chicken, E. (2005). Block-dependent thresholding in wavelet regression. J. Nonparametr. Statist. 17 467–491.
  • Chicken, E. and Cai, T. (2005). Block thresholding for density estimation: Local and global adaptivity. J. Multivariate Anal. 95 76–106.
  • Daubechies, I. (1992). Ten Lectures on Wavelets. SIAM, Philadelphia, PA.
  • DeVore, R. and Lorentz, G. G. (1993). Constructive Approximation. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 303. Springer, Berlin.
  • Donoho, D. L. and Johnstone, I. M. (1994a). Ideal spatial adaptation by wavelet shrinkage. Biometrika 81 425–455.
  • Donoho, D. L. and Johnstone, I. M. (1994b). Minimax risk over lp-balls for lq-error. Probab. Theory Related Fields 99 277–303.
  • Donoho, D. L. and Johnstone, I. M. (1995). Adapt 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., Liu, R. C. and MacGibbon, B. (1990). Minimax risk over hyperrectangles, and implications. Ann. Statist. 18 1416–1437.
  • Efromovich, S. Y. (1985). Nonparametric estimation of a density of unknown smoothness. Theor. Probab. Appl. 30 557–661.
  • Gao, H.-Y. (1998). Wavelet shrinkage denoising using the nonnegative garrote. J. Comput. Graph. Statist. 7 469–488.
  • Hall, P., Kerkyacharian, G. and Picard, D. (1998). Block threshold rules for curve estimation using kernel and wavelet methods. Ann. Statist. 26 922–942.
  • Hall, P., Kerkyacharian, G. and Picard, D. (1999). On the minimax optimality of block thresholded wavelet estimators. Statist. Sinica 9 33–50.
  • Johnstone, I. M. (1999). Wavelet shrinkage for correlated data and inverse problems: Adaptivity results. Statist. Sinica 9 51–84.
  • Johnstone, I. M. (2002). Function estimation and Gaussian sequence model. Unpublished manuscript.
  • Johnstone, I. M. and Silverman, B. W. (2005). Empirical Bayes selection of wavelet thresholds. Ann. Statist. 33 1700–1752.
  • Kerkyacharian, G., Picard, D. and Tribouley, K. (1996). Lp adaptive density estimation. Bernoulli 2 229–247.
  • Meyer, Y. (1992). Wavelets and Operators. Cambridge Studies in Advanced Mathematics 37. Cambridge Univ. Press, Cambridge. Translated from the 1990 French original by D. H. Salinger.
  • Stein, C. (1981). Estimation of the mean of a multivariate normal distribution. Ann. Statist. 9 1135–1151.
  • Strang, G. (1992). Wavelet and dilation equations: A brief introduction. SIAM Rev. 31 614–627.
  • Triebel, H. (1983). Theory of Function Spaces. Monographs in Math. 78. Birkhäuser, Basel.