## Electronic Journal of Statistics

### Normalized and standard Dantzig estimators: Two approaches

#### Abstract

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

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

Dates
First available in Project Euclid: 22 June 2015

https://projecteuclid.org/euclid.ejs/1434988476

Digital Object Identifier
doi:10.1214/15-EJS1040

Mathematical Reviews number (MathSciNet)
MR3358327

Zentralblatt MATH identifier
1327.62408

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

#### Citation

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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