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Parallelism, uniqueness, and large‐sample asymptotics for the Dantzig selector
Author(s) -
Dicker Lee,
Lin Xihong
Publication year - 2013
Publication title -
canadian journal of statistics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.804
H-Index - 51
eISSN - 1708-945X
pISSN - 0319-5724
DOI - 10.1002/cjs.11151
Subject(s) - uniqueness , mathematics , estimator , linear regression , distribution (mathematics) , lasso (programming language) , convergence (economics) , asymptotic distribution , statistics , mathematical analysis , computer science , world wide web , economics , economic growth
The Dantzig selector (Candès & Tao, 2007) is a popular $\ell^{1}$ ‐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 the uniqueness of lasso estimators (Tibshirani, 1996). Large sample asymptotics for the Dantzig selector, that is, 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 & Fu, 2000), our uniqueness results are valid for an arbitrary number of predictors and observations. The Canadian Journal of Statistics 41: 23–35; 2013 © 2012 Statistical Society of Canada