Penalized estimators for non linear inverse problems

¹ Institut de Mathématiques, Équipe de Statistique et Probabilités, Université Paul Sabatier, 31000 Toulouse, France
² Departamento de Matemáticas, IVIC, Venezuela

Corresponding authors: loubes@math.univ-toulouse.fr cludena@ivic.ve

Received: 29 May 2006
Revised: 9 October 2007
Revised: 16 March 2008

Abstract

In this article we tackle the problem of inverse non linear ill-posed problems from a statistical point of view. We discuss the problem of estimating an indirectly observed function, without prior knowledge of its regularity, based on noisy observations. For this we consider two approaches: one based on the Tikhonov regularization procedure, and another one based on model selection methods for both ordered and non ordered subsets. In each case we prove consistency of the estimators and show that their rate of convergence is optimal for the given estimation procedure.