Limit Distribution of a Risk Estimate in the Problem of Inverting Linear Homogeneous Operators with a Random Sample Size

Abstract

Inverse statistical problems associated with the inversion of some linear homogeneous transformation arise in areas such as tomography, plasma physics, optics, etc. As a rule, noise is present in observations, and it is necessary to apply some regularization methods. Recently, methods of threshold processing of wavelet expansion coefficients have become popular. When using threshold processing methods, it is usually assumed that the number of decomposition coefficients is fixed and the noise distribution is Gaussian. This model has been well studied in the literature, and the optimal threshold values have been calculated for different classes of signal functions. However, in some situations, the sample size is not known in advance and has to be modeled by some random variable. In this paper, a model with a random number of observations is considered, and it is shown that the limit distribution of the mean-square risk estimate is a shift-scale mixture of normal laws.