Perturbation bootstrap in adaptive Lasso
自适应 Lasso 的扰动自助法AI 解读 ↓
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自适应 Lasso 的扰动自助法
- 研究问题
- 用于自适应 Lasso 的扰动自助法是否具有二阶正确性，如何改进其分布近似，并允许模型维度随样本量增长？
- 方法与思路
- 作者修改扰动自助法的目标函数，并对修改后的估计量作适当学生化处理，再通过理论、模拟和真实数据示例考察推断。
- 材料陈述的贡献
- 摘要称，原始扰动方案不具二阶正确性，而修改方案在允许维度增长时可达二阶正确性，并报告较好的有限样本表现。
- 阅读关注点
- 应核对目标函数修改、学生化方式、维度增长限制、惩罚参数与信号条件，以及二阶正确性的误差定义和模拟比较设计。
- 阅读边界
- 仅据版本未注明的 OpenAlex 索引摘要；摘要未列出定理条件或数值结果，所称精度改进不能推广至任意高维或弱信号情形。
原文摘要
The Adaptive Lasso (Alasso) was proposed by Zou [J. Amer. Statist. Assoc. 101 (2006) 1418–1429] as a modification of the Lasso for the purpose of simultaneous variable selection and estimation of the parameters in a linear regression model. Zou [J. Amer. Statist. Assoc. 101 (2006) 1418–1429] established that the Alasso estimator is variable-selection consistent as well as asymptotically Normal in the indices corresponding to the nonzero regression coefficients in certain fixed-dimensional settings. In an influential paper, Minnier, Tian and Cai [J. Amer. Statist. Assoc. 106 (2011) 1371–1382] proposed a perturbation bootstrap method and established its distributional consistency for the Alasso estimator in the fixed-dimensional setting. In this paper, however, we show that this (naive) perturbation bootstrap fails to achieve second-order correctness in approximating the distribution of the Alasso estimator. We propose a modification to the perturbation bootstrap objective function and show that a suitably Studentized version of our modified perturbation bootstrap Alasso estimator achieves second-order correctness even when the dimension of the model is allowed to grow to infinity with the sample size. As a consequence, inferences based on the modified perturbation bootstrap will be more accurate than the inferences based on the oracle Normal approximation. We give simulation studies demonstrating good finite-sample properties of our modified perturbation bootstrap method as well as an illustration of our method on a real data set.