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February 2014 Adaptive robust variable selection
Jianqing Fan, Yingying Fan, Emre Barut
Ann. Statist. 42(1): 324-351 (February 2014). DOI: 10.1214/13-AOS1191

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.

Citation

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Jianqing Fan. Yingying Fan. Emre Barut. "Adaptive robust variable selection." Ann. Statist. 42 (1) 324 - 351, February 2014. https://doi.org/10.1214/13-AOS1191

Information

Published: February 2014
First available in Project Euclid: 19 March 2014

zbMATH: 1296.62144
MathSciNet: MR3189488
Digital Object Identifier: 10.1214/13-AOS1191

Subjects:
Primary: 62J07
Secondary: 62H12

Keywords: Adaptive weighted $L_{1}$ , high dimensions , oracle properties , robust regularization

Rights: Copyright © 2014 Institute of Mathematical Statistics

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Vol.42 • No. 1 • February 2014
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