The mean square risk of nonlinear regularization in the problem of inversion of linear homogeneous operators with a random sample size

The problems of constructing estimates from observations, which represent a linear transformation of the initial data, arise in many application areas, such as computed tomography, optics, plasma physics, and gas dynamics. In the presence of noise in the observations, as a rule, it is necessary to apply regularization methods. Recently, the methods of threshold processing of wavelet expansion coefficients have become popular. This is explained by the fact that such methods are simple, computationally efficient, and have the ability to adapt to functions which have different degrees of regularity at different areas. The analysis of errors of these methods is an important practical task, since it allows assessing the quality of both the methods themselves and the equipment used. When using threshold processing methods, it is usually assumed that the number of expansion coefficients is fixed and the noise distribution is Gaussian. This model is well studied in literature and optimal threshold values are 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 a random variable. In this paper, the author considers a model with a random number of observations containing Gaussian noise and estimates the order of the mean-square risk with an increasing sample size.