Electronic Journal of Statistics

Normalized and standard Dantzig estimators: Two approaches

Jan Mielniczuk and Hubert Szymanowski

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We reconsider the definition of the Dantzig estimator and show that, in contrast to the LASSO, standardization of an experimental matrix leads in general to a different estimator than in the case when it is based on the original data. The properties of the first method, resulting in what is called here the normalized Dantzig estimator are studied and the results on its estimation and prediction error are compared with similar results for the standard version. It is shown that in general the normalized version yields tighter estimation and prediction bounds than the other approach. In the correct specification case tighter bounds are obtained for the normalized Dantzig estimator than for the LASSO. Numerical examples indicate that in the case of imbalanced data the normalized estimator also performs better than the standard version.

Article information

Electron. J. Statist., Volume 9, Number 1 (2015), 1335-1356.

Received: October 2013
First available in Project Euclid: 22 June 2015

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62J05: Linear regression 62J07: Ridge regression; shrinkage estimators
Secondary: 90C25: Convex programming

Linear model high dimensionality Dantzig selector LASSO normalization constrained optimization Karush-Kuhn-Tucker conditions


Mielniczuk, Jan; Szymanowski, Hubert. Normalized and standard Dantzig estimators: Two approaches. Electron. J. Statist. 9 (2015), no. 1, 1335--1356. doi:10.1214/15-EJS1040. https://projecteuclid.org/euclid.ejs/1434988476

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