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The Dantzig selector (Candes and Tao, 2007) is a popular l1-regularizationmethod for variable selection and estimation in linear regression. We present avery weak geometric condition on the observed predictors which is related toparallelism and, when satisfied, ensures the uniqueness of Dantzig selectorestimators. The condition holds with probability 1, if the predictors are drawnfrom a continuous distribution. We discuss the necessity of this condition foruniqueness and also provide a closely related condition which ensuresuniqueness of lasso estimators (Tibshirani, 1996). Large sample asymptotics forthe Dantzig selector, i.e. almost sure convergence and the asymptoticdistribution, follow directly from our uniqueness results and a continuityargument. The limiting distribution of the Dantzig selector is generallynon-normal. Though our asymptotic results require that the number of predictorsis fixed (similar to (Knight and Fu, 2000)), our uniqueness results are validfor an arbitrary number of predictors and observations.

Parallelism, Uniqueness, and Large-Sample Asymptotics for the Dantzig Selector