The Penalized Analytic Center Estimator
In a linear regression model, the Dantzig selector (Candès and Tao, 2007 ) minimizes the L 1 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...
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| Published in | Econometric reviews Vol. 35; no. 8-10; pp. 1471 - 1484 |
|---|---|
| Main Author | |
| Format | Journal Article |
| Language | English |
| Published |
New York
Taylor & Francis
25.11.2016
Taylor & Francis Ltd |
| Subjects | |
| Online Access | Get full text |
| ISSN | 0747-4938 1532-4168 |
| DOI | 10.1080/07474938.2015.1092800 |
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| Abstract | In a linear regression model, the Dantzig selector (Candès and Tao,
2007
) minimizes the L
1
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 L
1
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/λ
2
. |
|---|---|
| AbstractList | In a linear regression model, the Dantzig selector (Candès and Tao, 2007) 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. In a linear regression model, the Dantzig selector (Candes and Tao, 2007) minimizes the L1 norm of the regression coefficients subject to a bound λ on the Lâ%z 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. In a linear regression model, the Dantzig selector (Candes and Tao, 2007) minimizes the L sub(1) norm of the regression coefficients subject to a bound lambda on the L sub( infinity ) 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 lambda and an additional tuning parameter r) minimize objective functions that are the sum of the L sub(1) 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 lambda and, when lambda is sufficiently large, the estimators depend approximately on r and lambda via r/ lambda super(2). In a linear regression model, the Dantzig selector (Candès and Tao, 2007 ) minimizes the L 1 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 L 1 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/λ 2 . |
| Author | Knight, Keith |
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| Cites_doi | 10.1214/07-AOAS131 10.1111/j.2517-6161.1990.tb01786.x 10.1111/j.1467-9868.2005.00503.x 10.1002/cjs.11151 10.1002/cem.1180010105 10.1145/800057.808695 10.1137/1.9781611971316 10.1007/s10107-007-0170-0 10.1214/08-AOS620 10.1214/009053606000001523 10.1214/009053607000000433 10.1214/08-AOS625 10.1111/j.1467-9868.2008.00668.x 10.1111/j.1467-9868.2011.00771.x 10.1214/aos/1015957397 10.1111/j.2517-6161.1996.tb02080.x 10.1080/07350015.2013.836104 10.1080/00401706.1970.10488634 |
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| Snippet | In a linear regression model, the Dantzig selector (Candès and Tao,
2007
) minimizes the L
1
norm of the regression coefficients subject to a bound λ on the L... In a linear regression model, the Dantzig selector (Candès and Tao, 2007) minimizes the L1 norm of the regression coefficients subject to a bound λ on the L∞... In a linear regression model, the Dantzig selector (Candes and Tao, 2007) minimizes the L sub(1) norm of the regression coefficients subject to a bound lambda... In a linear regression model, the Dantzig selector (Candes and Tao, 2007) minimizes the L1 norm of the regression coefficients subject to a bound λ on the Lâ%z... |
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| SubjectTerms | Analytic center C210 Dantzig selector Econometrics Economic models Estimators Lasso Mathematical analysis Mathematical models Norms Regression Regression analysis Regression coefficients Selectors Shrinkage estimation Tuning |
| Title | The Penalized Analytic Center Estimator |
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