ABSTRACT INVERSE ESTIMATION WITH APPLICATION TO DECONVOLUTION ON LOCALLY COMPACT ABELIAN GROUPS

ARNOUD C. M. VAN ROOIJ^{1} AND FRITS H. RUYMGAART^{2}

1Department of Mathematics, Katholieke Universiteit Nijmegen, 6525 ED Nijmegen,

The Netherlands

2Department of Mathematics and Statistics, Texas Tech University, Lubbock, TX 79409, U.S.A.

(Received December 22, 1997; revised June 19, 2000)

Abstract. Recovery of the unknown parameter in an abstract inverse estimation model can be based on regularizing the inverse of the operator defining the model. Such regularized-inverse type estimators are constructed with the help of a version of the spectral theorem due to Halmos, after suitable preconditioning. A lower bound to the minimax risk is obtained exploiting the van Trees inequality. The proposed estimators are shown to be asymptotically optimal in the sense that their risk converges to zero, as the sample size tends to infinity, at the same rate as this lower bound. The general theory is applied to deconvolution on locally compact Abelian groups, including both indirect density and indirect regression function estimation.

Key words and phrases: Abstract inverse estimation, indirect curve estimation, illposed problem, regularized-inverse type estimator, locally compact Abelian group, deconvolution.