  We propose to address the common problem of linear estimation in linear
statistical models by using a model selection approach via penalization.
Depending then on the framework in which the linear statistical model is
considered namely the regression framework or the inverse problem framework, a
data-driven model selection criterion is obtained either under general
assumptions, or under the mild assumption of model identifiability
respectively. The proposed approach was stimulated by the important recent
non-asymptotic model selection results due to Birg\'e and Massart mainly (Birge
and Massart 2007), and our results in this paper, like theirs, are
non-asymptotic and turn to be sharp.
  Our main contribution in this paper resides in the fact that these linear
estimators are not necessarily least-squares estimators but can be any linear
estimators. The proposed approach finds therefore potential applications in
countless fields of engineering and applied science (image science, signal
processing,applied statistics, coding, to name a few) in which one is
interested in recovering some unknown vector quantity of interest as the one,
for example, which achieves the best trade-off between a term of fidelity to
data, and a term of regularity or/and parsimony of the solution. The proposed
approach provides then such applications with an interesting model selection
framework that allows them to achieve such a goal.
