Risk estimate for the block thresholding method of solving inverse statistical problems with data on a random grid

O. V. Shestakov<sup>abc

a</sup> Department of Mathematical Statistics, Faculty of Computational Mathematics and Cybernetics, M. V. Lomo- nosov Moscow State University, 1-52 Leninskie Gory, GSP-1, Moscow 119991, Russian Federation
^b Federal Research Center “Computer Science and Control” of the Russian Academy of Sciences, 44-2 Vavilov Str., Moscow 119333, Russian Federation
^c Moscow Center for Fundamental and Applied Mathematics, M. V. Lomonosov Moscow State University, 1 Leninskie Gory, GSP-1, Moscow 119991, Russian Federation

Abstract: Wavelet analysis methods are widely used to solve inverse statistical problems for inverting linear homogeneous operators. The advantage of these methods is their computational efficiency and the ability to adapt to both the operator type and local features of the estimated function. To suppress the noise in the observed data, threshold processing of the expansion coefficients of the observed function over the wavelet basis is used. One of the most effective is the block thresholding method in which the expansion coefficients are processed in groups that allows taking into account information about neighboring coefficients. Sometimes, the nature of the data is such that observations are recorded at random times. If the sample points form a variation series constructed from a uniform distribution sample over the data recording interval, then the use of threshold processing procedures is adequate and does not worsen the quality of the estimates obtained. The paper analyzes the estimate of the mean square risk of the block thresholding method and shows that under certain conditions, this estimate is strongly consistent and asymptotically normal.

Keywords: linear homogeneous operator, wavelets, block thresholding, unbiased risk estimate, random samples.

Received: 22.06.2025
Accepted: 15.08.2025

DOI: 10.14357/19922264250303