We consider estimating multitask quantile regression under the transnormal model, with focus on high-dimensional setting. We derive a surprisingly simple closed-form solution through rank-based covariance regularization. In particular, we propose the rank-based ℓ1 penalization with positive-definite constraints for estimating sparse covariance matrices, and the rank-based banded Cholesky decomposition regularization for estimating banded precision matrices. By taking advantage of the alternating direction method of multipliers, nearest correlation matrix projection is introduced that inherits sampling properties of the unprojected one. Our work combines strengths of quantile regression and rank-based covariance regularization to simultaneously deal with nonlinearity and nonnormality for high-dimensional regression. Furthermore, the proposed method strikes a good balance between robustness and efficiency, achieves the “oracle”-like convergence rate, and provides the provable prediction interval under the high-dimensional setting. The finite-sample performance of the proposed method is also examined. The performance of our proposed rank-based method is demonstrated in a real application to analyze the protein mass spectroscopy data. Supplementary materials for this article are available online.
Bibliographical noteFunding Information:
The authors sincerely thank the associate editor and referees for their help comments and suggestions. Jianqing Fan’s research is supported in part by R01GM100474-04 and National Science Foundation grants DMS-1206464 and DMS-1406266. Lingzhou Xue’s research is supported by the National Institutes of Health grant R01-GM072611-09 and National Science Foundation grant DMS-1505256. Hui Zou’s research is supported by NSF grants DMS-0846068 and DMS-1505111.
© 2016 American Statistical Association.
- Alternating direction method of multipliers
- Cholesky decomposition
- Copula model
- Optimal transformation
- Prediction interval
- Quantile regression
- Rank correlation