Informatics and Applications

2025, Volume 19, Issue 2, pp 9-16

UNBIASED RISK ESTIMATE FOR THE FIRM SHRINKAGE METHOD OF SOLVING LINEAR INVERSE PROBLEMS

  • O. V Shestakov

Abstract

Inverse statistical problems arise in such areas as astronomy, plasma physics, computational tomography, etc. In this case, the observed data usually contain noise and, therefore, it is necessary to apply noise suppression methods. In situations where the problem is related to the inversion of a linear homogeneous operator, noise suppression methods based on the wavelet transform and thresholding procedures have proven themselves to be effective. These methods are computationally efficient and adapt well to local features of signals. The most common types of thresholding are hard and soft thresholding. However, hard thresholding produces estimates with a large variance, while soft thresholding introduces additional bias. In an attempt to get rid of these drawbacks, various alternative types of thresholding have been proposed in recent years. This paper considers a thresholding procedure with two thresholds, which behaves like soft thresholding for small values of wavelet coefficients and like hard thresholding for large values. For this type of thresholding, an unbiased estimate of the mean-square risk is constructed and its statistical properties are analyzed. An algorithm for calculating the threshold that minimizes this estimate is also described.

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