TY - JOUR

T1 - Asymptotically minimax and Bayes estimation in a deconvolution problem

AU - Ermakov, M.

N1 - Copyright: Copyright 2008 Elsevier B.V., All rights reserved.

PY - 2003/12

Y1 - 2003/12

N2 - We consider a deconvolution problem with a random noise. The noise is a product of a Gaussian stationary random process by a weight function εh ∈ L2(R1) with constant ε > 0. We study the asymptotically minimax and Bayes settings (ε → 0). In the minimax model a priori information is given that the solution belongs to a ball in Sobolev space W2β(R1). For such a priori information we find an asymptotically minimax estimator of the solution. In the Bayes setting the noise is the same. The solution is a realization of a random process defined as a product of a Gaussian stationary random process by a weight function h1 ∈ L2(R1). We show that the standard Wiener filters remain asymptotically Bayes estimators for this modification of Wiener filtration. The introduction of weight functions h, h1 ∈ L2(R1) is the main difference from the standard settings. This allows us not to make the traditional assumptions that the powers of noise and solutions are infinite or tend to infinity.

AB - We consider a deconvolution problem with a random noise. The noise is a product of a Gaussian stationary random process by a weight function εh ∈ L2(R1) with constant ε > 0. We study the asymptotically minimax and Bayes settings (ε → 0). In the minimax model a priori information is given that the solution belongs to a ball in Sobolev space W2β(R1). For such a priori information we find an asymptotically minimax estimator of the solution. In the Bayes setting the noise is the same. The solution is a realization of a random process defined as a product of a Gaussian stationary random process by a weight function h1 ∈ L2(R1). We show that the standard Wiener filters remain asymptotically Bayes estimators for this modification of Wiener filtration. The introduction of weight functions h, h1 ∈ L2(R1) is the main difference from the standard settings. This allows us not to make the traditional assumptions that the powers of noise and solutions are infinite or tend to infinity.

UR - http://www.scopus.com/inward/record.url?scp=0346946895&partnerID=8YFLogxK

U2 - 10.1088/0266-5611/19/6/007

DO - 10.1088/0266-5611/19/6/007

M3 - Article

AN - SCOPUS:0346946895

VL - 19

SP - 1339

EP - 1359

JO - Inverse Problems

JF - Inverse Problems

SN - 0266-5611

IS - 6

ER -