  The Huber's criterion is a useful method for robust regression. The adaptive
least absolute shrinkage and selection operator (lasso) is a popular technique
for simultaneous estimation and variable selection. In the case of small sample
size and large covariables numbers, this penalty is not very satisfactory
variable selection method. In this paper, we introduce an adaptive reversed
version of Huber's criterion as a penalty function. We call this penalty
adaptive Berhu penalty. As for elastic net penalty, small coefficients
contribute their $\ell_1$ norm to this penalty while larger coefficients cause
it to grow quadratically (as ridge regression). We show that the estimator
associated with criterion such that ordinary least square or Huber's one
combining with adaptive Berhu penalty enjoys the oracle properties. In
addition, this procedure encourages a grouping effect. This approach is
compared with adaptive elastic net regularization. Extensive simulation studies
demonstrate satisfactory finite-sample performance of such procedure. A real
example is analyzed for illustration purposes.
  Keywords : Adaptive Berhu penalty; concomitant scale; elastic net penalty;
Huber's criterion; oracle property; robust estimation.
