Incorporating Graphical Structure of Predictors in Sparse Quantile Regression

Abstract

Quantile regression in high-dimensional settings is useful in analyzing high-dimensional heterogeneous data. In this article, different from existing methods in quantile regression which treat all the predictors equally with the same priori, we take advantage of the graphical structure among predictors to improve the performance of parameter estimation, model selection, and prediction in sparse quantile regression. It is shown under mild conditions that the proposed method enjoys the model selection consistency and the oracle properties. An alternating direction method of multipliers algorithm with a linearization technique is proposed to implement the proposed method numerically, and its convergence is justified. Simulation studies are conducted, showing that the proposed method is superior to existing methods in terms of estimation accuracy and predictive power. The proposed method is also applied to a real dataset.

KEYWORDS:

• ADMM algorithm
• Graphical structure
• Quantile regression
• Sparse regression

Supplementary Materials

The supplementary materials contain the proofs of Theorems 1–4 and some additional simulation results.

Acknowledgments

The authors are indebted to the editor, the associate editor, and two anonymous reviewers for their professional review and insightful comments that lead to significant improvements in the article.

Additional information

Funding

Zhanfeng Wang’s research is supported by the National Natural Science Foundation of China (grant no. 11971457) and Anhui Provincial Natural Science Foundation (no. 1908085MA06). Xianhui Liu’s research is supported by the National Natural Science Foundation of China (grant nos. 11701235, 11961028, 11971208, 61973145) and the Natural Science Foundation of Jiangxi Province (grant no. 2018ACB21002). Yuanyuan Lin’s research is supported by the Hong Kong Research Grants Council (grant no. 14311916 and 14306219), the National Natural Science Foundation of China (grant no. 71874028), and Direct Grants for Research, The Chinese University of Hong Kong.