The Annals of Statistics

Wavelet thresholding for non-necessarily Gaussian noise: idealism

R. Averkamp and C. Houdré

Full-text: Open access

Abstract

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.

Article information

Source
Ann. Statist., Volume 31, Number 1 (2003), 110-151.

Dates
First available in Project Euclid: 26 February 2003

Permanent link to this document
https://projecteuclid.org/euclid.aos/1046294459

Digital Object Identifier
doi:10.1214/aos/1046294459

Mathematical Reviews number (MathSciNet)
MR1962501

Zentralblatt MATH identifier
1102.62329

Subjects
Primary: 62G07: Density estimation 62C20: Minimax procedures
Secondary: 60G70: Extreme value theory; extremal processes 41A25: Rate of convergence, degree of approximation

Keywords
Wavelets thresholding minimax

Citation

Averkamp, R.; Houdré, C. Wavelet thresholding for non-necessarily Gaussian noise: idealism. Ann. Statist. 31 (2003), no. 1, 110--151. doi:10.1214/aos/1046294459. https://projecteuclid.org/euclid.aos/1046294459


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References

  • [1] ANTONIADIS, A. (1996). Smoothing noisy data with tapered coiflets series. Scand. J. Statist. 23 313-330.
  • [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,
  • PQ, 1996). CRM Proc. Lecture Notes 18 347-354. AMS, Providence, RI.
  • [3] AVERKAMP, R. and HOUDRÉ, C. (1999). Wavelet thresholding for non(necessarily) Gaussian noise: Functionality. Preprint. Available at http://www.math.gatech.edu/ houdre/
  • [4] BAKIROV, N. K. (1989). Extrema of the distributions of quadratic forms of Gaussian variables. Theory Probab. Appl. 34 207-215.
  • [5] 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.
  • [6] BRUCE, A. and GAO, H. Y. (1995). Wave shrink with semisoft shrinkage. Statist. Sci. Research Report 39.
  • [7] 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.
  • [8] DAUBECHIES, I. (1992). Ten Lectures on Wavelets. SIAM, Philadelphia.
  • [9] DELy ON, B. and JUDITSKY, A. (1995). Estimating wavelet coefficients. Wavelets and Statistics. Lecture Notes in Statist. 103 151-168. Springer, Berlin.
  • [10] DELy ON, B. and JUDITSKY, A. (1996). On minimax wavelet estimators. Appl. Comput. Harmon. Anal. 3 215-228.
  • [11] 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.
  • [12] DONOHO, D. L. (1995). De-noising by soft-thresholding. IEEE Trans. Inform. Theory 41 613- 627.
  • [13] DONOHO, D. L. and JOHNSTONE, I. M. (1994). Ideal spatial adaptation by wavelet shrinkage. Biometrika 81 425-455.
  • [14] DONOHO, D. L. and JOHNSTONE, I. M. (1995). Adapting to unknown smoothness via wavelet shrinkage. J. Amer. Statist. Assoc. 90 1200-1224.
  • [15] DONOHO, D. L. and JOHNSTONE, I. M. (1996). Neo-classical minimax problems, thresholding and adaptive function estimation. Bernoulli 2 39-62.
  • [16] DONOHO, D. L., JOHNSTONE, I. M., KERKy ACHARIAN, G. and PICARD, D. (1996). Density estimation by wavelet thresholding. Ann. Statist. 24 508-539.
  • [17] 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.
  • [18] FELLER, W. (1966). An Introduction to Probability Theory and Its Applications 2. Wiley, New York.
  • [19] GAO, H. Y. (1993). Wavelet estimation of spectral densities in time series analysis. Ph.D. dissertation, Univ. California, Berkeley.
  • [20] GAO, H. Y. and BRUCE, A. G. (1997). WaveShrink with firm shrinkage. Statist. Sinica 7 855- 874.
  • [21] HÄRDLE, W., KERKy ACHARIAN, G., PICARD, D. and TSy BAKOV, A. (1998). Wavelets, Approximation and Statistical Applications. Lecture Notes in Statist. 129. Springer, Berlin.
  • [22] 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.
  • [23] 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.
  • [24] KERKy ACHARIAN, G. and PICARD, D. (1992). Density estimation in Besov spaces. Statist. Probab. Lett. 13 15-24.
  • [25] KERKy ACHARIAN, G. and PICARD, D. (1993). Density estimation by kernel and wavelets methods: Optimality of Besov spaces. Statist. Probab. Lett. 18 327-336.
  • [26] KOLACZy K, E. D. (1997). Nonparametric estimation of gamma-ray burst intensities using Haar wavelets. Astrophysical J. 483 340-349.
  • [27] KOLACZy K, E. D. (1999). Wavelet shrinkage estimation of certain Poisson intensity signals using corrected thresholds. Statist. Sinica 9 119-136.
  • [28] 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.
  • [29] LEDOUX, M. AND TALAGRAND, M. (1991). Probability in Banach Spaces. Springer, Berlin.
  • [30] MARSHALL, A. W. and OLKIN, I. (1979). Inequalities: Theory of Majorization and Its Applications. Academic Press, New York.
  • [31] MEy ER, Y. (1990). Ondelettes et Opérateurs 1. Ondelettes. Paris, Hermann.
  • [32] MEy ER, Y. (1993). Wavelets, Algorithms and Applications. SIAM, Philadelphia.
  • [33] NEUMANN, M. H. and SPOKOINY, V. G. (1995). On the efficiency of wavelet estimators under arbitrary error distributions. Math. Methods Statist. 4 137-166.
  • [34] PETROV, V. V. (1995). Limit Theorems of Probability Theory. Clarendon Press, Oxford.
  • [35] SHORACK, G. R. and WELLNER, J. A. (1986). Empirical Processes with Applications to Statistics. Wiley, New York.
  • [36] STOUT, W. (1974). Almost Sure Convergence. Academic Press, New York.
  • [37] STRASSER, H. (1985). Mathematical Theory of Statistics. de Gruy ter, Berlin.
  • [38] VIDAKOVIC, B. (1999). Statistical Modeling by Wavelets. Wiley, New York.
  • [39] WANG, Y. (1996). Function estimation via wavelet shrinkage for long-memory data. Ann. Statist. 24 466-484.
  • ATLANTA, GEORGIA 30332 E-MAIL: houdre@math.gatech.edu