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

Non-asymptotic model selection for linear non least-squares estimation in regression models and inverse problems