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The Huber's criterion is a useful method for robust regression. The adaptiveleast absolute shrinkage and selection operator (lasso) is a popular techniquefor simultaneous estimation and variable selection. In the case of small samplesize and large covariables numbers, this penalty is not very satisfactoryvariable selection method. In this paper, we introduce an adaptive reversedversion of Huber's criterion as a penalty function. We call this penaltyadaptive Berhu penalty. As for elastic net penalty, small coefficientscontribute their $\ell_1$ norm to this penalty while larger coefficients causeit to grow quadratically (as ridge regression). We show that the estimatorassociated with criterion such that ordinary least square or Huber's onecombining with adaptive Berhu penalty enjoys the oracle properties. Inaddition, this procedure encourages a grouping effect. This approach iscompared with adaptive elastic net regularization. Extensive simulation studiesdemonstrate satisfactory finite-sample performance of such procedure. A realexample is analyzed for illustration purposes.  Keywords : Adaptive Berhu penalty; concomitant scale; elastic net penalty;Huber's criterion; oracle property; robust estimation.

The BerHu penalty and the grouped effect