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Limit distribution of a risk estimate using the vaguelette-wavelet decomposition of signals in a model with correlated noise

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Abstract

A signal function after the application of a linear homogenous operator in a model with correlated noise is estimated. The asymptotic properties of a risk estimate with thresholding of the coefficients of the vaguelette-wavelet decomposition of a signal are considered. Conditions for the asymptotic normality of the risk estimate are proposed.

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References

  1. D. Donoho and I. M. Johnstone, “Adapting to unknown smoothness via wavelet shrinkage,” J. Am. Stat. Assoc. 90, 1200–1224 (1995).

    Article  MATH  MathSciNet  Google Scholar 

  2. D. Donoho and I. M. Johnstone, “Ideal spatial adaptation via wavelet shrinkage,” Biometrika 81, 425–455 (1994).

    Article  MATH  MathSciNet  Google Scholar 

  3. D. J. Donoho, I. M. Johnstone, G. Kerkyacharian, and D. Picard, “Wavelet shrinkage: asymptopia?,” J. R. Statist. Soc. Ser. B 57, 301–369 (1995).

    MATH  MathSciNet  Google Scholar 

  4. J. S. Marron, S. Adak, I. M. Johnstone, M. H. Neumann, and P. Patil, “Exact risk analysis of wavelet regression,” J. Comput. Graph. Stat., 7, 278–309 (1998).

    Google Scholar 

  5. A. Antoniadis and J. Fan, “Regularization of wavelet approximations,” J. Am. Stat. Assoc. 96(455), 939–967 (2001).

    Article  MATH  MathSciNet  Google Scholar 

  6. A. V. Markin, “Limit distribution of the risk estimate with thresholding of wavelet coefficients,” inf. Primen. 3(4), 57–63 (2009).

    Google Scholar 

  7. A. V. Markin and O. V. Shestakov, “Consistency of risk estimation with thresholding of wavelet coefficients,” Moscow Univ. Comput. Math. Cybern. 34, 22–30 (2010).

    Article  MATH  MathSciNet  Google Scholar 

  8. O. V. Shestakov, “Approximation of the distribution of the risk estimate in thresholding of wavelet coefficients with the use of the normal distribution in the case of application of sample variance,” inf. Primen. 4(4), 73–81 (2010)

    MathSciNet  Google Scholar 

  9. O. V. Shestakov, “On the accuracy of approximation of the distribution of the risk estimate in thresholding of wavelet coefficients with the use of the normal distribution for an unknown level of noise,” Sist. Sredstva Inf., 22(1), 142–152 (2012).

    Google Scholar 

  10. O. V. Shestakov, “Asymptotic normality of adaptive wavelet thresholding risk estimation,” Dokl. Math. 86, 556–558 (2012).

    Article  MATH  MathSciNet  Google Scholar 

  11. O. V. Shestakov, “Dependence of the limit distribution of the risk estimate in thresholding of wavelet coefficients of a signal on type of the estimate of the noise variance in selection of the adaptive threshold,” T-Comm.-Telekomm. Transport, No. 1, 46–51 (2012).

    Google Scholar 

  12. O. V. Shestakov, “Central limit theorem for the function of generalized cross-validation in thresholding of wavelet coefficients,” inf. Primen. 7(2), 40–49 (2013).

    Google Scholar 

  13. A. A. Eroshenko and O. V. Shestakov, “Asymptotic characteristics of the risk estimate in thresholding of wavelet coefficients in a model with correlated noise,” inf. Primen. 8(1), 36–44 (2014).

    Google Scholar 

  14. A. A. Eroshenko and O. V. Shestakov, “Asymptotic normality of estimating risk upon the wavelet-vaguelette decomposition of a signal function in a model with correlated noise,” Moscow Univ. Comput. Math. Cybern. 38, 110–117 (2014).

    Article  Google Scholar 

  15. F. Abramovich and B. W. Silverman, “Wavelet decomposition approaches to statistical inverse problems,” Biometrika 85, 115–129 (1998).

    Article  MATH  MathSciNet  Google Scholar 

  16. I. Daubechies, Ten Lectures on Wavelets (SIAM, Philadelphia, 1992; NITS RKHD, Izhevsk-Moscow, 2001).

    Book  MATH  Google Scholar 

  17. S. A. Mallat, Wavelet Tour of Signal Processing (Academic, New York, 1999).

    MATH  Google Scholar 

  18. A. Boffess and F. A. Narkowich, First Course in Wavelets with Fourier Analysis (Prentice Hall, Upper Saddle River, 2001).

    Google Scholar 

  19. M. S. Taqqu, “Weak convergence to fractional Brownian morion and to the Rosenblatt process,” Z. Wahrscheinlichkeitstheor. Verwandte Geb. 31, 287–302 (1975).

    Article  MATH  MathSciNet  Google Scholar 

  20. I. M. Johnstone and B. W. Silverman, “Wavelet threshold estimates for data with correlated noise,” J. Roy. Statist. Soc. Ser. B, 59, 319–351 (1997).

    Article  MATH  MathSciNet  Google Scholar 

  21. N. Lee, PhD Thesis (Purdue Univ., Lafayette, 1997).

  22. R. C. Bradley, “Basic properties of strong mixing conditions. A survey and some open questions,” Probab. Surv. 2, 107–144 (2005).

    Article  MATH  MathSciNet  Google Scholar 

  23. M. Perligrad, “On the asymptotic normality of sequences of weak dependent random variables,” J. Theor. Prob. 9, 703–715 (1996).

    Article  Google Scholar 

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Authors and Affiliations

  1. Department of Computational Mathematics and Cybernetics, Moscow State University, Moscow, 119991, Russia

    A. A. Eroshenko, A. A. Kudryavtsev & O. V. Shestakov

  2. Institute for Problems of Informatics, Russian Academy of Sciences, Moscow, 117333, Russia

    O. V. Shestakov

Authors
  1. A. A. Eroshenko
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  2. A. A. Kudryavtsev
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  3. O. V. Shestakov
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Corresponding author

Correspondence to A. A. Eroshenko.

Additional information

Original Russian Text © A.A. Eroshenko, A.A. Kudryavtsev, O.V. Shestakov, 2015, published in Vestnik Moskovskogo Universiteta. Vychislitel’naya Matematika i Kibernetika, 2015, No. 1, pp. 12–18.

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Eroshenko, A.A., Kudryavtsev, A.A. & Shestakov, O.V. Limit distribution of a risk estimate using the vaguelette-wavelet decomposition of signals in a model with correlated noise. MoscowUniv.Comput.Math.Cybern. 39, 6–13 (2015). https://doi.org/10.3103/S0278641915010021

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  • DOI: https://doi.org/10.3103/S0278641915010021

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