Normalized and standard Dantzig estimators: Two approaches

Abstract

We reconsider the definition of the Dantzig estimator and show that, in contrast to the LASSO, standardization of an experimental matrix leads in general to a different estimator than in the case when it is based on the original data. The properties of the first method, resulting in what is called here the normalized Dantzig estimator are studied and the results on its estimation and prediction error are compared with similar results for the standard version. It is shown that in general the normalized version yields tighter estimation and prediction bounds than the other approach. In the correct specification case tighter bounds are obtained for the normalized Dantzig estimator than for the LASSO. Numerical examples indicate that in the case of imbalanced data the normalized estimator also performs better than the standard version.