## The Annals of Statistics

### Globally adaptive quantile regression with ultra-high dimensional data

#### 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.

#### Article information

Source
Ann. Statist., Volume 43, Number 5 (2015), 2225-2258.

Dates
Revised: April 2015
First available in Project Euclid: 16 September 2015

https://projecteuclid.org/euclid.aos/1442364151

Digital Object Identifier
doi:10.1214/15-AOS1340

Mathematical Reviews number (MathSciNet)
MR3396984

Zentralblatt MATH identifier
1327.62424

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

#### Citation

Zheng, Qi; Peng, Limin; He, Xuming. Globally adaptive quantile regression with ultra-high dimensional data. Ann. Statist. 43 (2015), no. 5, 2225--2258. doi:10.1214/15-AOS1340. https://projecteuclid.org/euclid.aos/1442364151

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#### Supplemental materials

• Supplement to “Globally adaptive quantile regression with ultra-high dimensional data”. Due to space constraints, additional simulation results, the proofs of technical lemmas and corollaries, justification for the proposed grid approximation, and sample codes are relegated to the supplement [Zheng, Peng and He (2015)].