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Wavelet Thresholding Risk Estimate for the Model with Random Samples and Correlated Noise

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  • Oleg Shestakov

    (Faculty of Computational Mathematics and Cybernetics, M. V. Lomonosov Moscow State University, Moscow 119991, Russia
    Institute of Informatics Problems, Federal Research Center “Computer Science and Control” of the Russian Academy of Sciences, Moscow 119333, Russia)

Abstract

Signal de-noising methods based on threshold processing of wavelet decomposition coefficients have become popular due to their simplicity, speed, and ability to adapt to signal functions with spatially inhomogeneous smoothness. The analysis of the errors of these methods is an important practical task, since it makes it possible to evaluate the quality of both methods and equipment used for processing. Sometimes the nature of the signal is such that its samples are recorded at random times. If the sample points form a variational series based on a sample from the uniform distribution on the data registration interval, then the use of the standard threshold processing procedure is adequate. The paper considers a model of a signal that is registered at random times and contains noise with long-term dependence. The asymptotic normality and strong consistency properties of the mean-square thresholding risk estimator are proved. The obtained results make it possible to construct asymptotic confidence intervals for threshold processing errors using only the observed data.

Suggested Citation

  • Oleg Shestakov, 2020. "Wavelet Thresholding Risk Estimate for the Model with Random Samples and Correlated Noise," Mathematics, MDPI, vol. 8(3), pages 1-8, March.
    • Handle: RePEc:gam:jmathe:v:8:y:2020:i:3:p:377-:d:329900
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    References listed on IDEAS

    as
    1. Antoniadis A. & Fan J., 2001. "Regularization of Wavelet Approximations," Journal of the American Statistical Association, American Statistical Association, vol. 96, pages 939-967, September.
    2. Cai, T. Tony & Brown, Lawrence D., 1999. "Wavelet estimation for samples with random uniform design," Statistics & Probability Letters, Elsevier, vol. 42(3), pages 313-321, April.
    3. Iain M. Johnstone & Bernard W. Silverman, 1997. "Wavelet Threshold Estimators for Data with Correlated Noise," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 59(2), pages 319-351.
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