High-dimensional quantile regression: Convolution smoothing and concave regularization

• 成果类型: Article
• 署名作者: Tan, Kean Ming; Wang, Lan; Zhou, Wen-Xin
• 署名单位: University of Michigan System; University of Michigan; University of Miami; University of California System; University of California San Diego
• 刊物名称: JOURNAL OF THE ROYAL STATISTICAL SOCIETY SERIES B-STATISTICAL METHODOLOGY
• ISSN/ISSBN: 1369-7412
• DOI: 10.1111/rssb.12485
• 发表日期: 2022
• 页码: 205-233
• 关键词: nonconcave penalized likelihood; variable selection; REGRESSION SHRINKAGE; M-ESTIMATORS; Consistency; Lasso; optimality

摘要

l(1)-penalized quantile regression (QR) is widely used for analysing high-dimensional data with heterogeneity. It is now recognized that the l(1)-penalty introduces non-negligible estimation bias, while a proper use of concave regularization may lead to estimators with refined convergence rates and oracle properties as the signal strengthens. Although folded concave penalized M-estimation with strongly convex loss functions have been well studied, the extant literature on QR is relatively silent. The main difficulty is that the quantile loss is piecewise linear: it is non-smooth and has curvature concentrated at a single point. To overcome the lack of smoothness and strong convexity, we propose and study a convolution-type smoothed QR with iteratively reweighted l(1)-regularization. The resulting smoothed empirical loss is twice continuously differentiable and (provably) locally strongly convex with high probability. We show that the iteratively reweighted l(1)-penalized smoothed QR estimator, after a few iterations, achieves the optimal rate of convergence, and moreover, the oracle rate and the strong oracle property under an almost necessary and sufficient minimum signal strength condition. Extensive numerical studies corroborate our theoretical results.