Parallelism, uniqueness, and large‐sample asymptotics for the Dantzig selector

Abstract

The Dantzig selector (Candès and Tao, 2007) is a popular ℓ1-regularization method for variable selection and estimation in linear regression. We present a very weak geometric condition on the observed predictors which is related to parallelism and, when satisfied, ensures the uniqueness of Dantzig selector estimators. The condition holds with probability 1, if the predictors are drawn from a continuous distribution. We discuss the necessity of this condition for uniqueness and also provide a closely related condition which ensures uniqueness of lasso estimators (Tibshirani, 1996). Large sample asymptotics for the Dantzig selector, i.e. almost sure convergence and the asymptotic distribution, follow directly from our uniqueness results and a continuity argument. The limiting distribution of the Dantzig selector is generally non-normal. Though our asymptotic results require that the number of predictors is fixed (similar to (Knight and Fu, 2000)), our uniqueness results are valid for an arbitrary number of predictors and observations.