Sparse Composite Quantile Regression in Ultrahigh Dimensions with Tuning Parameter Calibration

Yuwen Gu, Hui Zou

Research output: Contribution to journalArticlepeer-review

22 Scopus citations

Abstract

When estimating coefficients in a linear model, the (sparse) composite quantile regression was first proposed in Zou and Yuan (2008) as an efficient alternative to the (sparse) least squares to handle arbitrary error distribution. The highly nonsmooth nature of the composite loss in the sparse composite quantile regression makes its theoretical analysis as well as numerical computation much more challenging than the least squares method. The theory in Zou and Yuan (2008) was proven under fixed-dimension asymptotics and the estimator was computed via linear programming that does not scale well with high dimensions. In this paper, we study the sparse composite quantile regression under ultrahigh dimensionality and make three contributions. First, we provide a non-asymptotic analysis of both the lasso and the folded concave penalized composite quantile regression, which reveals a practical way of achieving the oracle estimator. Second, we construct a novel information criterion for selecting the regularization parameter in the folded concave penalized composite quantile regression and prove its selection consistency. Third, we exploit the structure of the composite loss and design a specialized optimization algorithm for computing the penalized composite quantile regression via the alternating direction method of multipliers. We conduct extensive simulations to illustrate the theoretical results. Our analysis provides a unified treatment of the concentration inequalities involving the composite loss. Those inequalities could be of independent interest.

Original languageEnglish (US)
Article number9112219
Pages (from-to)7132-7154
Number of pages23
JournalIEEE Transactions on Information Theory
Volume66
Issue number11
DOIs
StatePublished - Nov 2020

Bibliographical note

Publisher Copyright:
© 1963-2012 IEEE.

Keywords

  • Composite quantile regression
  • information criterion
  • sparsity
  • ultrahigh-dimensional data

Fingerprint

Dive into the research topics of 'Sparse Composite Quantile Regression in Ultrahigh Dimensions with Tuning Parameter Calibration'. Together they form a unique fingerprint.

Cite this