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Ideal denoising within a family of tree-structured wavelet estimators. (English) Zbl 1274.62228

Summary: 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.

MSC:

62G05 Nonparametric estimation
62G20 Asymptotic properties of nonparametric inference
65T60 Numerical methods for wavelets

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References:

[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. · Zbl 1092.62005 · doi:10.1214/009053606000000074
[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. · Zbl 1320.62064 · doi:10.1214/08-EJS205
[5] Autin, F. (2008). Maxisets for, \mu -thresholding Rules. Test , 17 (2), 332-349. · Zbl 1196.62033 · doi:10.1007/s11749-006-0035-5
[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. · Zbl 1086.62043 · doi:10.1214/009053605000000471
[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. · Zbl 0954.62047 · doi:10.1214/aos/1018031262
[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. · Zbl 0992.65151 · doi:10.1006/acha.2001.0336
[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. · Zbl 0997.62025 · doi:10.1006/acha.2000.0333
[12] Daubechies, I. (1992)., Ten Lectures on Wavelets . SIAM, Philadelphia. · Zbl 0776.42018
[13] Donoho, D.L., and Johnstone, I.M. (1994). Ideal Spatial Adaptation by Wavelet Shrinkage., Biometrika , 81 (3), 425-455. · Zbl 0815.62019 · doi:10.1093/biomet/81.3.425
[14] Donoho, D.L. (1997). CART and Best-ortho-basis., Annals of Statistics , 25 (5), 1870-1911. · Zbl 0942.62044 · doi:10.1214/aos/1069362377
[15] Engel, J. (1994). A simple Wavelet Approach to Nonparametric Regression from Recursive Partitioning Schemes., Journal of Multivariate Analysis , 49 (2), 242-254. · Zbl 0795.62034 · doi:10.1006/jmva.1994.1024
[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. · Zbl 1392.62282 · doi:10.1198/jasa.2010.tm09132
[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. · Zbl 0899.62002
[19] Jansen, M. (2001). Noise Reduction by Wavelet Thresholding. Springer Verlag, Lecture Notes in Statistics, vol. 161, 224, pp. · Zbl 0989.94001 · doi:10.1007/978-1-4613-0145-5
[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. · Zbl 0993.62034 · doi:10.1111/1467-842X.00205
[21] Kerkyacharian, G., and Picard, D. (2000). Thresholding Algorithms, Maxisets and Well Concentrated Bases., Test , 9 (2), 283-344. · Zbl 1107.62323 · doi:10.1007/BF02595738
[22] Kerkyacharian, G., and Picard, D. (2002). Minimax or maxisets?, Bernoulli , 8 (2), 219-253. · Zbl 1006.62005
[23] Shapiro, J. (1993). Embedded image coding using zero trees of wavelet coefficients., IEEE Transactions on Signal Processing , 41 (12), 3445-3462. · Zbl 0841.94020 · doi:10.1109/78.258085
[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. · Zbl 1069.68594 · doi:10.1016/j.patcog.2003.11.006
[25] Tsybakov, A. (2008). Introduction to Nonparametric Estimation. Springer Series in Statistics, 214, pp. · Zbl 1176.62032
[26] Vidakovic, B. (1999). Statistical Modelling by Wavelets, John Wiley & Sons, Inc., New York, 384, pp. · Zbl 0924.62032
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