The Generalized SURE Estimate for Threshold Processing when Solving Inverse Statistical Problems

A wide range of scientific problems are related to ‘‘indirect’’ measurements containing noise. The researcher is interested in obtaining an estimate of some object $$f$$ , but the data are available only on the transformation $$Kf$$ , where $$K$$ is some linear operator, and $$Kf$$ is additionally distorted by noise. Reconstruction of $$f$$ from such indirect observations using classical singular value decomposition often leads to unsatisfactory results if the original function $$f$$ is spatially inhomogeneous. As an alternative, methods based on wavelet decompositions that take these features into account have been proposed. Error analysis of these methods is an important practical task, since it allows one to evaluate the quality of both the methods themselves and the equipment used. The paper considers threshold processing methods for wavelet decomposition coefficients and investigates the statistical properties of this method. Conditions are given under which strong consistency and asymptotic normality of the unbiased mean square risk estimate in the problem of linear homogeneous operator inversion take place.