The Penalized Analytic Center Estimator

In a linear regression model, the Dantzig selector (Candès and Tao, 2007 Candès, E., Tao, T. (2007). The Dantzig selector: Statistical estimation when p is much larger than n. Annals of Statistics 35:2313–2351.Crossref], Web of Science ®] , Google Scholar]) minimizes the L₁ norm of the regression coefficients subject to a bound λ on the L_∞ norm of the covariances between the predictors and the residuals; the resulting estimator is the solution of a linear program, which may be nonunique or unstable. We propose a regularized alternative to the Dantzig selector. These estimators (which depend on λ and an additional tuning parameter r) minimize objective functions that are the sum of the L₁ norm of the regression coefficients plus r times the logarithmic potential function of the Dantzig selector constraints, and can be viewed as penalized analytic centers of the latter constraints. The tuning parameter r controls the smoothness of the estimators as functions of λ and, when λ is sufficiently large, the estimators depend approximately on r and λ via r/λ².