Ultra-high dimensional data often display heterogeneity due to either heteroscedastic variance or other forms of non-location-scale covariate effects. To accommodate heterogeneity, we advocate a more general interpretation of sparsity, which assumes that only a small number of covariates influence the conditional distribution of the response variable, given all candidate covariates; however, the sets of relevant covariates may differ when we consider different segments of the conditional distribution. In this framework, we investigate the methodology and theory of nonconvex, penalized quantile regression in ultra-high dimension. The proposed approach has two distinctive features: (1) It enables us to explore the entire conditional distribution of the response variable, given the ultra-high-dimensional covariates, and provides a more realistic picture of the sparsity pattern; (2) it requires substantially weaker conditions compared with alternative methods in the literature; thus, it greatly alleviates the difficulty of model checking in the ultra-high dimension. In theoretic development, it is challenging to deal with both the nonsmooth loss function and the nonconvex penalty function in ultra-high-dimensional parameter space. We introduce a novel, sufficient optimality condition that relies on a convex differencing representation of the penalized loss function and the subdifferential calculus. Exploring this optimality condition enables us to establish the oracle property for sparse quantile regression in the ultra-high dimension under relaxed conditions. The proposed method greatly enhances existing tools for ultra-high-dimensional data analysis. Monte Carlo simulations demonstrate the usefulness of the proposed procedure. The real data example we analyzed demonstrates that the new approach reveals substantially more information as compared with alternative methods. This article has online supplementary material.
Bibliographical noteFunding Information:
Lan Wang is Associate Professor, School of Statistics, University of Minnesota, Minneapolis, MN 55455 (E-mail: firstname.lastname@example.org). Yichao Wu is Assistant Professor, Department of Statistics, North Carolina State University, Raleigh, NC 27695 (E-mail: email@example.com). Runze Li is the corresponding author and Professor, Department of Statistics and the Methodology Center, the Pennsylvania State University, University Park, PA 16802-2111 (E-mail: firstname.lastname@example.org). Lan Wang’s research was supported by an NSF grant DMS1007603. Yichao Wu’s research was supported by an NSF grant DMS-0905561 and 1055210 and an NIH/NCI grant R01 CA-149569. Runze Li’s research was supported by an NSF grant DMS 0348869 and grants from NNSF of China, 11028103 and 10911120395, and NIDA, NIH grants R21 DA024260 and P50 DA10075. The content is solely the responsibility of the authors and does not necessarily represent the official views of the NCI, the NIDA, or the NIH. The authors are indebted to the referees, the Associate Editor, and the Coeditor for their valuable comments, which have significantly improved the article.
- Penalized quantile regression
- Ultra-high-dimensional data