×

zbMATH — the first resource for mathematics

Asymptotic performance of projection estimators in standard and hyperbolic wavelet bases. (English) Zbl 1336.62125
Summary: We provide a novel treatment of the ability of the standard (wavelet-tensor) and of the hyperbolic (tensor product) wavelet bases to build nonparametric estimators of multivariate functions. First, we give new results about the limitations of wavelet estimators based on the standard wavelet basis regarding their inability to optimally reconstruct functions with anisotropic smoothness. Next, we provide optimal or near optimal rates at which both linear and nonlinear hyperbolic wavelet estimators are well-suited to reconstruct functions from anisotropic Besov spaces and subsequently we characterize the set of all the functions that are well reconstructed by these methods with respect to these rates. As a first main result, we furnish novel arguments to understand the primordial role of sparsity and thresholding in multivariate contexts, in particular by showing a stronger exposure of linear methods to the curse of dimensionality. Second, we propose an adaptation of the well known block thresholding method to a hyperbolic wavelet basis and show its ability to estimate functions with anisotropic smoothness at the optimal minimax rate. Therefore, we prove the pertinence of horizontal information pooling even in high dimensional settings. Numerical experiments illustrate the finite samples properties of the studied estimators.

MSC:
62H12 Estimation in multivariate analysis
62G08 Nonparametric regression and quantile regression
62G05 Nonparametric estimation
62G20 Asymptotic properties of nonparametric inference
42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
65T60 Numerical methods for wavelets
PDF BibTeX XML Cite
Full Text: DOI Euclid
References:
[1] Abry, P., Clausel, M., Jaffard, S., Roux, S. G., and Vedel, B. (2015). The hyperbolic wavelet transform: An efficient tool for multifractal analysis of anisotropic fields., Revista Matematica Iberoamericana , 32(1):313-348. · Zbl 1336.46028
[2] Abry, P., Clausel, M., Jaffard, S., and Vedel, B. (2013). The hyperbolic wavelet transform for self-similar anisotropic texture analysis., IEEE Transactions on Image Processing , 22(11):4353-4363. · Zbl 1373.94354
[3] Antoniadis, A., Bigot, J., and Sapatinas, J. (2001). Wavelet estimators in nonparametric regression: A comparative simulation study., Journal of Statistical Software , 6(6):1-83.
[4] Autin, F. (2008a). Maxisets for \(\mu\) thresholding rules., Test , 17:332-349. · Zbl 1196.62033
[5] Autin, F. (2008b). On the performances of a new thresholding procedure using tree structure., Electronic Journal of Statistics , 2:412-431. · Zbl 1320.62064
[6] Autin, F., Claeskens, G., and Freyermuth, J.-M. (2014a). Hyperbolic wavelet thresholding methods and the curse of dimensionality through the maxiset approach., Applied and Computational Harmonic Analysis , 36:239-255. · Zbl 1336.94013
[7] Autin, F., Freyermuth, J.-M., and von Sachs, R. (2011). Ideal denoising within a family of tree-structured wavelet estimators., Electronic Journal of Statistics , 5:829-855. · Zbl 1274.62228
[8] Autin, F., Freyermuth, J.-M., and von Sachs, R. (2012). Combining thresholding rules: A new way to improve the performance of wavelet estimators., Journal of Nonparametric Statistics , 24(4):905-922. · Zbl 1254.62032
[9] Autin, F., Freyermuth, J.-M., and von Sachs, R. (2014b). Block-threshold-adapted estimators via a maxiset approach., Scandinavian Journal of Statistics , 41:240-258. · Zbl 1349.62093
[10] Autin, F., Lepennec, E., and Tribouley, K. (2010). Thresholding methods to estimate the copula density., Journal of Multivariate Analysis , 101(1):200-222. · Zbl 1177.62075
[11] Benhaddou, R., Pensky, M., and Picard, D. (2013). Anisotropic de-noising in functional deconvolution model with dimension-free convergence rates., Electronic Journal of Statistics , 7:1686-1715. · Zbl 1294.62057
[12] Cai, T. (1999). Adaptive wavelet estimation: A block thresholding and oracle inequality approach., The Annals of Statistics , 27(3):898-924. · Zbl 0954.62047
[13] Cai, T. (2002). On block thresholding in wavelet regression: Adaptivity, block size, and threshold level., Statistica Sinica , 12:1241-1273. · Zbl 1004.62036
[14] Cai, T. (2008). On information pooling, adaptability and superefficiency in nonparametric function estimation., Journal of Multivariate Analysis , 99:421-436. · Zbl 1148.62020
[15] Cai, T. and Zhou, H. (2009). A data-driven block thresholding approach to wavelet estimation., The Annals of Statistics , 37:569-595. · Zbl 1162.62032
[16] 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
[17] Comminges, L. and Dalalyan, A. (2013). Minimax testing of a composite null hypothesis defined via a quadratic functional in the model of regression., IEEE Transactions on Signal Processing , 7:146-190. · Zbl 1337.62090
[18] Daubechies, I. (1992)., Ten Lectures on Wavelets . Number 61 in CBMS/NSF Series in Applied Math. SIAM, Philadelphia. · Zbl 0776.42018
[19] DeVore, R., Konyagin, S., and Temlyakov, V. (1998). Hyperbolic wavelet approximation., Constructive Approximation , 14:1-26. · Zbl 0895.41016
[20] Donoho, D. (1993). Unconditional bases are optimal bases for data compression and for statistical estimation., Applied and Computational Harmonic Analysis , 1:100-115. · Zbl 0796.62083
[21] Donoho, D. and Johnstone, I. (1994). Ideal spatial adaptation by wavelet shrinkage., Biometrika , 81(3):425-455. · Zbl 0815.62019
[22] Donoho, D., Johnstone, I. M., Kerkyacharian, G., and Picard, D. (1995). Wavelet shrinkage: Asymptopia?, Journal of The Royal Statistical Society, Series B , 57(2):301-337. · Zbl 0827.62035
[23] Duarte, M. and Baraniuk, R. (2012). Kronecker compressive sensing., IEEE Transaction on Image Processing , 21(2):494-504. · Zbl 1372.94379
[24] Hochmuth, R. (2002). \(n\)-term approximation in anisotropic function spaces., Mathematische Nachrichten , 244:131-149. · Zbl 1009.42024
[25] Ingster, Y., Laurent, B., and Marteau, C. (2014). Signal detection for inverse problems in a multidimensional framework., Mathematical Methods of Statistics , 23(4):279-305. · Zbl 1308.62091
[26] Ingster, Y. and Stepanova, N. (2011). Estimation and detection of functions from anisotropic sobolev classes., Electronic Journal of Statistics , 5:484-506. · Zbl 1274.62319
[27] Jansen, M. (2015)., ThreshLab: Matlab algorithms for wavelet noise reduction . Matlab package version 4.1.2.
[28] Lepski, O. (2014). Adaptive estimation over anisotropic functional classes via oracle approach. Technical, report. · Zbl 1328.62213
[29] Massart, P. (2007)., Concentration Inequalities and Model Selection . Ecole d’été de Probabilités de Saint-Flour 2003. Lecture Notes in Mathematics 1896. Springer, Berlin/Heidelberg. · Zbl 1170.60006
[30] Meyer, Y. (1990)., Ondelettes et Opérateurs . Actualités Mathématiques, Hermann, Paris. · Zbl 0694.41037
[31] Nason, G. (2013)., Wavethresh: Wavelets statistics and transforms . R package version 4.6.6.
[32] Neumann, M. (2000). Multivariate wavelet thresholding in anisotropic function spaces., Statistica Sinica , 10:399-431. · Zbl 0982.62039
[33] Neumann, M. and von Sachs, R. (1997). Wavelet thresholding in anisotropic function classes and application to adaptive estimation of evolutionary spectra., The Annals of Statistics , 25(1):38-76. · Zbl 0871.62081
[34] Reiss, M. (2008). Asymptotic equivalence for nonparametric regression with multivariate and random design., The Annals of Statistics , 36(4):1957-1982. · Zbl 1142.62023
[35] Temlyakov, V. (2002). Universal bases and greedy algorithms for anisotropic function classes., Constructive Approximation , 18:529-550. · Zbl 1025.41009
[36] Vidakovic, B. (1999)., Statistical Modelling by Wavelets . John Wiley & Sons, Inc., New York. · Zbl 0924.62032
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.