Model selection consistency of Dantzig selector

Consistency of model selection hinges on the correlation between significant and insignificant predictors for “large p, small n” problems. Thus, Irrepresentable Conditions play an important role in consistency, that insignificant predictors are irrepresentable by significant ones. In this paper, we provide Irrepresentable Conditions when the Dantzig selector is applied; they ensure that the Dantzig selector consistently selects the true model with fixed p and diverging p (number of predictors) even at an exponential rate of n. Our conditions are sufficient for a strong sign consistency and Weak Irrepresentable Conditions are necessary for a weak sign consistency. Strong sign consistency leads to the conventional consistency of the estimation. As a by-product, the results also show the difference between the Dantzig selector and the Lasso when consistency is at issoe. Simulation studies are performed to examine the theoretical results.