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October 2015 Globally adaptive quantile regression with ultra-high dimensional data
Qi Zheng, Limin Peng, Xuming He
Ann. Statist. 43(5): 2225-2258 (October 2015). DOI: 10.1214/15-AOS1340

Abstract

Quantile regression has become a valuable tool to analyze heterogeneous covaraite-response associations that are often encountered in practice. The development of quantile regression methodology for high-dimensional covariates primarily focuses on the examination of model sparsity at a single or multiple quantile levels, which are typically prespecified ad hoc by the users. The resulting models may be sensitive to the specific choices of the quantile levels, leading to difficulties in interpretation and erosion of confidence in the results. In this article, we propose a new penalization framework for quantile regression in the high-dimensional setting. We employ adaptive $L_{1}$ penalties, and more importantly, propose a uniform selector of the tuning parameter for a set of quantile levels to avoid some of the potential problems with model selection at individual quantile levels. Our proposed approach achieves consistent shrinkage of regression quantile estimates across a continuous range of quantiles levels, enhancing the flexibility and robustness of the existing penalized quantile regression methods. Our theoretical results include the oracle rate of uniform convergence and weak convergence of the parameter estimators. We also use numerical studies to confirm our theoretical findings and illustrate the practical utility of our proposal.

Citation

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Qi Zheng. Limin Peng. Xuming He. "Globally adaptive quantile regression with ultra-high dimensional data." Ann. Statist. 43 (5) 2225 - 2258, October 2015. https://doi.org/10.1214/15-AOS1340

Information

Received: 1 September 2014; Revised: 1 April 2015; Published: October 2015
First available in Project Euclid: 16 September 2015

zbMATH: 1327.62424
MathSciNet: MR3396984
Digital Object Identifier: 10.1214/15-AOS1340

Subjects:
Primary: 62J07
Secondary: 62H12

Keywords: adaptive penalized quantile regression , model selection oracle property , Ultra-high dimensional data , varying covariate effects

Rights: Copyright © 2015 Institute of Mathematical Statistics

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Vol.43 • No. 5 • October 2015
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