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.
Bibliographical noteFunding Information:
Manuscript received November 13, 2019; accepted May 26, 2020. Date of publication June 9, 2020; date of current version October 21, 2020. The work of Yuwen Gu was supported in part by the UConn Research Excellence Program. The work of Hui Zou was supported in part by the National Science Foundation (NSF) under Grant DMS-1915842. (Corresponding author: Hui Zou.) Yuwen Gu is with the Department of Statistics, University of Connecticut, Storrs, CT 06269 USA (e-mail: firstname.lastname@example.org).
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- Composite quantile regression
- information criterion
- ultrahigh-dimensional data