Electronic Journal of Statistics

Ideal denoising within a family of tree-structured wavelet estimators

Florent Autin, Jean-Marc Freyermuth, and Rainer von Sachs

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We focus on the performances of tree-structured wavelet estimators belonging to a large family of keep-or-kill rules, namely the Vertical Block Thresholding family. For each estimator, we provide the maximal functional space (maxiset) for which the quadratic risk reaches a given rate of convergence. Following a discussion on the maxiset embeddings, we identify the ideal estimator of this family, that is the one associated with the largest maxiset. We emphasize the importance of such a result since the ideal estimator is different from the usual (plug-in) estimator used to mimic the performances of the Oracle. Finally, we confirm the good performances of the ideal estimator compared to the other elements of that family through extensive numerical experiments.

Article information

Electron. J. Statist., Volume 5 (2011), 829-855.

First available in Project Euclid: 9 August 2011

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Zentralblatt MATH identifier

Primary: 62G05: Estimation 62G20: Asymptotic properties 41A25: Rate of convergence, degree of approximation 42C40: Wavelets and other special systems 65T60: Wavelets

Besov spaces curve estimation CART maxiset and oracle approaches rate of convergence thresholding methods tree structure wavelet estimators


Autin, Florent; Freyermuth, Jean-Marc; von Sachs, Rainer. Ideal denoising within a family of tree-structured wavelet estimators. Electron. J. Statist. 5 (2011), 829--855. doi:10.1214/11-EJS628. https://projecteuclid.org/euclid.ejs/1312906585

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  • [1] Abramovich, F., Benjamini, Y., Donoho, D., Johnstone, I. (2006). Adapting to Unknown Sparsity by Controlling the False Discovery Rate., Annals of Statistics, 34(2), 584-653.
  • [2] Antoniadis, A., Bigot, J., Sapatinas, T. (2001). Wavelet Estimators in Nonparametric Regression: a Comparative Simulation Study., Journal of Statistical Software, 6(6), 1-83.
  • [3] Autin, F. (2004). Maxiset Point of View in Nonparametric Estimation., Ph.D. at university of Paris 7 - France.
  • [4] Autin, F. (2008). On the Performances of a New Thresholding Procedure using Tree Structure., Electronic Journal of Statistics, 2, 412-431.
  • [5] Autin, F. (2008). Maxisets for, μ-thresholding Rules. Test, 17(2), 332-349.
  • [6] Autin, F., Picard, D., and Rivoirard, V. (2006). Large variance gaussian priors in Bayesian nonparametric estimation: a maxiset approach., Mathematical Methods of Statistics, 15(4), 349-373.
  • [7] Averkamp, R., Houdré (2005). Wavelet Thresholding for Non Necessarily Gaussian Noise: Functionality., Annals of Statistics, 33(5), 2164-2193.
  • [8] Baraniuk, R. (1999). Optimal Tree Approximation Using Wavelets., Proceedings of SPIE Conference on Wavelet Applications in Signal and Image Processing VII, Eds A. J. Aldroubi and M. Unser, Bellingham, WA:SPIE, 196-207.
  • [9] Cai, T. (1999). Adaptive Wavelet Estimation: a Block Thresholding and Oracle Inequality Approach., Annals of Statistics, 27(3), 898-924.
  • [10] Cohen, A., Dahmen W., Daubechies I., and DeVore, R. (2001). Tree Approximation and Optimal Encoding., Applied and Computational Harmonic Analysis, 11(2), 192-226.
  • [11] Cohen, A., De Vore, R., Kerkyacharian, G., and Picard, D. (2001). Maximal Spaces with Given Rate of Convergence for Thresholding Algorithms., Applied and Computational Harmonic Analysis, 11, 167-191.
  • [12] Daubechies, I. (1992)., Ten Lectures on Wavelets. SIAM, Philadelphia.
  • [13] Donoho, D.L., and Johnstone, I.M. (1994). Ideal Spatial Adaptation by Wavelet Shrinkage., Biometrika, 81(3), 425-455.
  • [14] Donoho, D.L. (1997). CART and Best-ortho-basis., Annals of Statistics, 25(5), 1870-1911.
  • [15] Engel, J. (1994). A simple Wavelet Approach to Nonparametric Regression from Recursive Partitioning Schemes., Journal of Multivariate Analysis, 49(2), 242-254.
  • [16] Engel, J. (1999). Tree Structured Estimation with Haar Wavelets. Verlag, 159, pp.
  • [17] Freyermuth, J.-M., Ombao, H., and von Sachs R. (2010). Tree-Structured Wavelet Estimation in a Mixed Effects Model for Spectra of Replicated Time Series., Journal of the American Statistical Association, 105(490), 634-646.
  • [18] Härdle, W. and Kerkycharian, G. and Picard D. and Tsybakov, A. (1998). Wavelets, approximation, and statistical applications. Springer Verlag, Lectures Notes in Statistics, vol., 129.
  • [19] Jansen, M. (2001). Noise Reduction by Wavelet Thresholding. Springer Verlag, Lecture Notes in Statistics, vol. 161, 224, pp.
  • [20] Lee, T. (2002). Tree based wavelet regression for correlated data using the minimum description length principle., Australian and New Zealand Journal of Statistics, 44(1), 23-39.
  • [21] Kerkyacharian, G., and Picard, D. (2000). Thresholding Algorithms, Maxisets and Well Concentrated Bases., Test, 9(2), 283-344.
  • [22] Kerkyacharian, G., and Picard, D. (2002). Minimax or maxisets?, Bernoulli, 8(2), 219-253.
  • [23] Shapiro, J. (1993). Embedded image coding using zero trees of wavelet coefficients., IEEE Transactions on Signal Processing, 41(12), 3445-3462.
  • [24] Sun, J., Gu, D., Chen, Y., and Zhang, S. (2004). A multiscale edge detection algorithm based on wavelet domain vector hidden markov tree model., Pattern Recognition, 37, 1315-1324.
  • [25] Tsybakov, A. (2008). Introduction to Nonparametric Estimation. Springer Series in Statistics, 214, pp.
  • [26] Vidakovic, B. (1999). Statistical Modelling by Wavelets, John Wiley & Sons, Inc., New York, 384, pp.