  We consider the problem of variables selection and estimation in linear
regression model in situations where the number of parameters diverges with the
sample size. We propose the adaptive Generalized Ridge-Lasso (\mbox{AdaGril})
which is an extension of the the adaptive Elastic Net. AdaGril incorporates
information redundancy among correlated variables for model selection and
estimation. It combines the strengths of the quadratic regularization and the
adaptively weighted Lasso shrinkage. In this paper, we highlight the grouped
selection property for AdaCnet method (one type of AdaGril) in the equal
correlation case. Under weak conditions, we establish the oracle property of
AdaGril which ensures the optimal large performance when the dimension is high.
Consequently, it achieves both goals of handling the problem of collinearity in
high dimension and enjoys the oracle property. Moreover, we show that AdaGril
estimator achieves a Sparsity Inequality, i. e., a bound in terms of the number
of non-zero components of the 'true' regression coefficient. This bound is
obtained under a similar weak Restricted Eigenvalue (RE) condition used for
Lasso. Simulations studies show that some particular cases of AdaGril
outperform its competitors.
