  For consistency (even oracle properties) of estimation and model prediction,
almost all existing methods of variable/feature selection critically depend on
sparsity of models. However, for ``large $p$ and small $n$" models sparsity
assumption is hard to check and particularly, when this assumption is violated,
the consistency of all existing estimations is usually impossible because
working models selected by existing methods such as the LASSO and the Dantzig
selector are usually biased. To attack this problem, we in this paper propose
adaptive post-Dantzig estimation and model prediction. Here the adaptability
means that the consistency based on the newly proposed method is adaptive to
non-sparsity of model, choice of shrinkage tuning parameter and dimension of
predictor vector. The idea is that after a sub-model as a working model is
determined by the Dantzig selector, we construct a globally unbiased sub-model
by choosing suitable instrumental variables and nonparametric adjustment. The
new estimation of the parameters in the sub-model can be of the asymptotic
normality. The consistent estimator, together with the selected sub-model and
adjusted model, improves model predictions. Simulation studies show that the
new approach has the significant improvement of estimation and prediction
accuracies over the Gaussian Dantzig selector and other classical methods have.
