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Averkamp, R.; Houdré, C.

Wavelet thresholding for non-necessarily Gaussian noise: idealism. (English) Zbl 1102.62329

Ann. Stat. 31, No. 1, 110-151 (2003).
Summary: For various types of noise (exponential, normal mixture, compactly supported, ...) wavelet thresholding methods are studied. Problems linked to the existence of optimal thresholds are tackled, and minimaxity properties of the methods also analyzed. A coefficient dependent method for choosing thresholds is also briefly presented.

Cited in 15 Documents

MSC:

62G07 Density estimation
42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems

Keywords:

wavelets; thresholding; minimax

Software:

S+WAVELETS
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\textit{R. Averkamp} and \textit{C. Houdré}, Ann. Stat. 31, No. 1, 110--151 (2003; Zbl 1102.62329)
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References:

[1] ANTONIADIS, A. (1996). Smoothing noisy data with tapered coiflets series. Scand. J. Statist. 23 313-330. · Zbl 0861.62028
[2] AVERKAMP, R. and HOUDRÉ, C. (1999). Wavelet thresholding for non(necessarily) Gaussian noise: A preliminary report. In Spline Functions and the Theory of Wavelets (Montréal, · Zbl 1179.62049
[3] PQ, 1996). CRM Proc. Lecture Notes 18 347-354. AMS, Providence, RI.
[4] AVERKAMP, R. and HOUDRÉ, C. (1999). Wavelet thresholding for non(necessarily) Gaussian noise: Functionality. Preprint. Available at http://www.math.gatech.edu/ houdre/ URL: · Zbl 1179.62049
[5] BAKIROV, N. K. (1989). Extrema of the distributions of quadratic forms of Gaussian variables. Theory Probab. Appl. 34 207-215. · Zbl 0692.60019
[6] BICKEL, P. (1983). Minimax estimation of the mean of a normal distribution subject to doing well at a point. In Recent Advances in Statistics (M. H. Rizvi, J. S. Rustagi and D. Siegmund, eds.) 511-528. Academic Press, New York. · Zbl 0545.62029
[7] BRUCE, A. and GAO, H. Y. (1995). Wave shrink with semisoft shrinkage. Statist. Sci. Research Report 39.
[8] CHAMBOLLE, A., DEVORE, R. A., LEE, N.-Y. and LUCIER, B. J. (1998). Nonlinear wavelet image processing: Variational problems, compression and noise removal through wavelet shrinkage. IEEE Trans. Image Process. 7 319-335. · Zbl 0993.94507
[9] DAUBECHIES, I. (1992). Ten Lectures on Wavelets. SIAM, Philadelphia. · Zbl 0776.42018
[10] DELy ON, B. and JUDITSKY, A. (1995). Estimating wavelet coefficients. Wavelets and Statistics. Lecture Notes in Statist. 103 151-168. Springer, Berlin. · Zbl 0832.62028
[11] DELy ON, B. and JUDITSKY, A. (1996). On minimax wavelet estimators. Appl. Comput. Harmon. Anal. 3 215-228. · Zbl 0865.62023
[12] DEVORE, R. A. and LUCIER, B. (1992). Fast wavelet techniques for near-optimal image processing. In 1992 IEEE Military Communications Conference 3 1129-1135. IEEE, New York.
[13] DONOHO, D. L. (1995). De-noising by soft-thresholding. IEEE Trans. Inform. Theory 41 613- 627. · Zbl 0820.62002
[14] DONOHO, D. L. and JOHNSTONE, I. M. (1994). Ideal spatial adaptation by wavelet shrinkage. Biometrika 81 425-455. JSTOR: · Zbl 0815.62019
[15] DONOHO, D. L. and JOHNSTONE, I. M. (1995). Adapting to unknown smoothness via wavelet shrinkage. J. Amer. Statist. Assoc. 90 1200-1224. JSTOR: · Zbl 0869.62024
[16] DONOHO, D. L. and JOHNSTONE, I. M. (1996). Neo-classical minimax problems, thresholding and adaptive function estimation. Bernoulli 2 39-62. · Zbl 0877.62035
[17] DONOHO, D. L., JOHNSTONE, I. M., KERKy ACHARIAN, G. and PICARD, D. (1996). Density estimation by wavelet thresholding. Ann. Statist. 24 508-539. · Zbl 0860.62032
[18] EFRON, B. and MORRIS, C. (1971). Limiting the risk of Bay es and empirical Bay es estimators. I. The Bay es case. J. Amer. Statist. Assoc. 66 807-815. JSTOR: · Zbl 0229.62003
[19] FELLER, W. (1966). An Introduction to Probability Theory and Its Applications 2. Wiley, New York. · Zbl 0138.10207
[20] GAO, H. Y. (1993). Wavelet estimation of spectral densities in time series analysis. Ph.D. dissertation, Univ. California, Berkeley.
[21] GAO, H. Y. and BRUCE, A. G. (1997). WaveShrink with firm shrinkage. Statist. Sinica 7 855- 874. · Zbl 1067.62529
[22] HÄRDLE, W., KERKy ACHARIAN, G., PICARD, D. and TSy BAKOV, A. (1998). Wavelets, Approximation and Statistical Applications. Lecture Notes in Statist. 129. Springer, Berlin.
[23] JOHNSTONE, I. M. and SILVERMAN, B. W. (1997). Wavelet threshold estimators for data with correlated noise. J. Roy. Statist. Soc. Ser. B 59 319-351. JSTOR: · Zbl 0886.62044
[24] KERKy ACHARIAN, G. and PICARD, D. (1992). Estimation de densité par méthode de noy au et d’ondelettes: Les liens entre la géometrie du noy au et les contraintes de régularité. C. R. Acad. Sci. Paris Sér. I 315 79-84. · Zbl 0749.62027
[25] KERKy ACHARIAN, G. and PICARD, D. (1992). Density estimation in Besov spaces. Statist. Probab. Lett. 13 15-24. · Zbl 0749.62026
[26] KERKy ACHARIAN, G. and PICARD, D. (1993). Density estimation by kernel and wavelets methods: Optimality of Besov spaces. Statist. Probab. Lett. 18 327-336. · Zbl 0793.62019
[27] KOLACZy K, E. D. (1997). Nonparametric estimation of gamma-ray burst intensities using Haar wavelets. Astrophysical J. 483 340-349.
[28] KOLACZy K, E. D. (1999). Wavelet shrinkage estimation of certain Poisson intensity signals using corrected thresholds. Statist. Sinica 9 119-136. · Zbl 0927.62081
[29] KOVAC, A. and SILVERMAN, B. W. (2000). Extending the scope of wavelet regression methods by coefficient-dependent thresholding. J. Amer. Statist. Assoc. 95 172-183.
[30] LEDOUX, M. AND TALAGRAND, M. (1991). Probability in Banach Spaces. Springer, Berlin. · Zbl 0748.60004
[31] MARSHALL, A. W. and OLKIN, I. (1979). Inequalities: Theory of Majorization and Its Applications. Academic Press, New York. · Zbl 0437.26007
[32] MEy ER, Y. (1990). Ondelettes et Opérateurs 1. Ondelettes. Paris, Hermann.
[33] MEy ER, Y. (1993). Wavelets, Algorithms and Applications. SIAM, Philadelphia.
[34] NEUMANN, M. H. and SPOKOINY, V. G. (1995). On the efficiency of wavelet estimators under arbitrary error distributions. Math. Methods Statist. 4 137-166. · Zbl 0845.62034
[35] PETROV, V. V. (1995). Limit Theorems of Probability Theory. Clarendon Press, Oxford. · Zbl 0826.60001
[36] SHORACK, G. R. and WELLNER, J. A. (1986). Empirical Processes with Applications to Statistics. Wiley, New York. · Zbl 1170.62365
[37] STOUT, W. (1974). Almost Sure Convergence. Academic Press, New York. · Zbl 0321.60022
[38] STRASSER, H. (1985). Mathematical Theory of Statistics. de Gruy ter, Berlin. · Zbl 0594.62017
[39] VIDAKOVIC, B. (1999). Statistical Modeling by Wavelets. Wiley, New York. · Zbl 0924.62032
[40] WANG, Y. (1996). Function estimation via wavelet shrinkage for long-memory data. Ann. Statist. 24 466-484. · Zbl 0859.62042
[41] ATLANTA, GEORGIA 30332 E-MAIL: houdre@math.gatech.edu
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