## The Annals of Statistics

### Adaptive robust variable selection

#### Abstract

Heavy-tailed high-dimensional data are commonly encountered in various scientific fields and pose great challenges to modern statistical analysis. A natural procedure to address this problem is to use penalized quantile regression with weighted $L_{1}$-penalty, called weighted robust Lasso (WR-Lasso), in which weights are introduced to ameliorate the bias problem induced by the $L_{1}$-penalty. In the ultra-high dimensional setting, where the dimensionality can grow exponentially with the sample size, we investigate the model selection oracle property and establish the asymptotic normality of the WR-Lasso. We show that only mild conditions on the model error distribution are needed. Our theoretical results also reveal that adaptive choice of the weight vector is essential for the WR-Lasso to enjoy these nice asymptotic properties. To make the WR-Lasso practically feasible, we propose a two-step procedure, called adaptive robust Lasso (AR-Lasso), in which the weight vector in the second step is constructed based on the $L_{1}$-penalized quantile regression estimate from the first step. This two-step procedure is justified theoretically to possess the oracle property and the asymptotic normality. Numerical studies demonstrate the favorable finite-sample performance of the AR-Lasso.

#### Article information

Source
Ann. Statist., Volume 42, Number 1 (2014), 324-351.

Dates
First available in Project Euclid: 19 March 2014

Permanent link to this document
https://projecteuclid.org/euclid.aos/1395234980

Digital Object Identifier
doi:10.1214/13-AOS1191

Mathematical Reviews number (MathSciNet)
MR3189488

Zentralblatt MATH identifier
1296.62144

Subjects
Primary: 62J07: Ridge regression; shrinkage estimators
Secondary: 62H12: Estimation

#### Citation

Fan, Jianqing; Fan, Yingying; Barut, Emre. Adaptive robust variable selection. Ann. Statist. 42 (2014), no. 1, 324--351. doi:10.1214/13-AOS1191. https://projecteuclid.org/euclid.aos/1395234980

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#### Supplemental materials

• Supplementary material: Supplementary material for: Adaptive robust variable selection. Due to space constraints, the proofs of Theorems 3 and 5 and the results of the real life data-set study are relegated to the supplement [Fan, Fan and Barut (2014)].