Adaptive robust variable selection

研究问题: 面对重尾的超高维数据，如何缓解惩罚估计偏差，使变量选择与参数推断在较弱误差分布条件下仍有理论保障？
方法与思路: 作者研究带加权 L1 惩罚的分位数回归，并提出两步 AR-Lasso，用第一步惩罚分位数估计构造第二步权重。
材料陈述的贡献: 摘要称在维度可随样本量指数增长的设置下，建立了 WR-Lasso 与两步 AR-Lasso 的预言性质和渐近正态性。
阅读关注点: 应核对稀疏性、设计矩阵和信号强度条件，自适应权重的构造要求，以及重尾误差允许范围和有限样本比较。
阅读边界: 以下沿用原解读对于其预印本摘要的限制：这是 arXiv 预印本，不能据此确认同行评审或期刊接收；摘要未列明所谓温和条件，也未提供数值研究的结果。复用旧预印本摘要解读；摘要与本条目相同，未另读刊稿全文，原请求未记录具体预印本版本。

GPT-6.1 Sol · 高 · 2026-10-03

原文摘要

Heavy-tailed high-dimensional data are commonly encountered in various scientific fields and pose great challenges to modern statistical analysis. A natural procedure to address this problem is to use penalized quantile regression with weighted $L_1$ -penalty, called weighted robust Lasso (WR-Lasso), in which weights are introduced to ameliorate the bias problem induced by the $L_1$ -penalty. In the ultra-high dimensional setting, where the dimensionality can grow exponentially with the sample size, we investigate the model selection oracle property and establish the asymptotic normality of the WR-Lasso. We show that only mild conditions on the model error distribution are needed. Our theoretical results also reveal that adaptive choice of the weight vector is essential for the WR-Lasso to enjoy these nice asymptotic properties. To make the WR-Lasso practically feasible, we propose a two-step procedure, called adaptive robust Lasso (AR-Lasso), in which the weight vector in the second step is constructed based on the $L_1$ -penalized quantile regression estimate from the first step. This two-step procedure is justified theoretically to possess the oracle property and the asymptotic normality. Numerical studies demonstrate the favorable finite-sample performance of the AR-Lasso.Heavy-tailed high-dimensional data are commonly encountered in various scientific fields and pose great challenges to modern statistical analysis. A natural procedure to address this problem is to use penalized quantile regression with weighted $L_1$-penalty, called weighted robust Lasso (WR-Lasso), in which weights are introduced to ameliorate the bias problem induced by the $L_1$-penalty. In the ultra-high dimensional setting, where the dimensionality can grow exponentially with the sample size, we investigate the model selection oracle property and establish the asymptotic normality of the WR-Lasso. We show that only mild conditions on the model error distribution are needed. Our theoretical results also reveal that adaptive choice of the weight vector is essential for the WR-Lasso to enjoy these nice asymptotic properties. To make the WR-Lasso practically feasible, we propose a two-step procedure, called adaptive robust Lasso (AR-Lasso), in which the weight vector in the second step is constructed based on the $L_1$-penalized quantile regression estimate from the first step. This two-step procedure is justified theoretically to possess the oracle property and the asymptotic normality. Numerical studies demonstrate the favorable finite-sample performance of the AR-Lasso.

摘要来源：arXiv 1205.4795v3 · 预印本版本摘要

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