Sparse Convoluted Rank Regression in High Dimensions

Abstract

Wang et al. studied the high-dimensional sparse penalized rank regression and established its nice theoretical properties. Compared with the least squares, rank regression can have a substantial gain in estimation efficiency while maintaining a minimal relative efficiency of $86.4\%$. However, the computation of penalized rank regression can be very challenging for high-dimensional data, due to the highly nonsmooth rank regression loss. In this work we view the rank regression loss as a nonsmooth empirical counterpart of a population level quantity, and a smooth empirical counterpart is derived by substituting a kernel density estimator for the true distribution in the expectation calculation. This view leads to the convoluted rank regression loss and consequently the sparse penalized convoluted rank regression (CRR) for high-dimensional data. We prove some interesting asymptotic properties of CRR. Under the same key assumptions for sparse rank regression, we establish the rate of convergence of the ${\mathcal{l}}_1$-penalized CRR for a tuning free penalization parameter and prove the strong oracle property of the folded concave penalized CRR. We further propose a high-dimensional Bayesian information criterion for selecting the penalization parameter in folded concave penalized CRR and prove its selection consistency. We derive an efficient algorithm for solving sparse convoluted rank regression that scales well with high dimensions. Numerical examples demonstrate the promising performance of the sparse convoluted rank regression over the sparse rank regression. Our theoretical and numerical results suggest that sparse convoluted rank regression enjoys the best of both sparse least squares regression and sparse rank regression. Supplementary materials for this article are available online.

KEYWORD:

• Convolution
• Efficiency
• High dimensions
• Information criterion
• Rank regression

Supplementary Materials

All the technical proofs and additional simulation results are relegated to the supplementary file.

Acknowledgments

The authors would like to thank the Associate Editor and referees for their helpful comments and suggestions that greatly improved the quality of this article. Zou’s research is supported in part by NSF grants 1915842 and 2015120.

Disclosure Statement

The authors report there are no competing interests to declare.

Additional information

Funding

The authors would like to thank the Associate Editor and referees for their helpful comments and suggestions that greatly improved the quality of this article. Zou’s research is supported in part by NSF grants 1915842 and 2015120.