Journal of Mathematical Sciences

, Volume 237, Issue 6, pp 826–830 | Cite as

Averaged Probability of the Error in Calculating Wavelet Coefficients for the Random Sample Size

  • O. V. ShestakovEmail author

Signal denoising methods based on the threshold processing of wavelet coefficients are widely used in various application areas. When applying these methods, it is usually assumed that the number of wavelet coefficients is fixed, and the noise distribution is Gaussian. Such a model has been well studied in the literature, and optimal threshold values have been calculated for different signal classes and loss functions. However, in some situations the sample size is not known in advance and is modeled by a random variable. In this paper, we consider a model with a random number of observations contaminated by a Gaussian noise, and study the behavior of the loss function based on the probabilities of errors in calculating wavelet coefficients for a growing sample size.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    D. Donoho and I. M. Johnstone, “Ideal spatial adaptation via wavelet shrinkage,” Biometrika, 81, No. 3, 425–455 (1994).MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    D. Donoho and I. M. Johnstone, “Minimax estimation via wavelet shrinkage,” Ann. Stat., 26, No. 3, 879–921 (1998).MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    M. Jansen, Noise Reduction by Wavelet Thresholding, Springer, New York (2001).CrossRefzbMATHGoogle Scholar
  4. 4.
    J. Sadasiva, S. Mukherjee, and C. S. Seelamantula, “An optimum shrinkage estimator based on minimum-probability-of-error criterion and application to signal denoising,” in: 2014 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), Piscataway, New Jersey (2014), pp. 4249–4253.Google Scholar
  5. 5.
    A. A. Kudryavtsev and O. V. Shestakov, “Asymptotic behavior of the threshold minimizing the average probability of error in calculation of wavelet coefficients,” Dokl. Math., 93, No. 3, 295–299 (2016).MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    A. A. Kudryavtsev and O. V. Shestakov, “Asymptotically optimal wavelet thresholding in the models with non-Gaussian noise distributions,” Dokl. Math., 94, No. 3, 615–619 (2016).MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    S. Mallat, A Wavelet Tour of Signal Processing, Academic Press, New York (1999).zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Mathematical Statistics, Faculty of Computational Mathematics and CyberneticsM.V. Lomonosov Moscow State UniversityMoscowRussia
  2. 2.Institute of Informatics ProblemsFederal Research Center “Computer Science and Control” of the Russian Academy of SciencesMoscowRussia

Personalised recommendations