Dai Linlin, Chen Kani, Sun Zhihua, Liu Zhenqiu, Li Gang
Southwestern University of Finance and Economics, Chengdu, China.
Department of Mathematics, Hong Kong University of Science and Technology, Hong Kong.
J Multivar Anal. 2018 Nov;168:334-351. doi: 10.1016/j.jmva.2018.08.007. Epub 2018 Aug 23.
This paper studies the asymptotic properties of a sparse linear regression estimator, referred to as broken adaptive ridge (BAR) estimator, resulting from an -based iteratively reweighted penalization algorithm using the ridge estimator as its initial value. We show that the BAR estimator is consistent for variable selection and has an oracle property for parameter estimation. Moreover, we show that the BAR estimator possesses a grouping effect: highly correlated covariates are naturally grouped together, which is a desirable property not known for other oracle variable selection methods. Lastly, we combine BAR with a sparsity-restricted least squares estimator and give conditions under which the resulting two-stage sparse regression method is selection and estimation consistent in addition to having the grouping property in high- or ultrahigh-dimensional settings. Numerical studies are conducted to investigate and illustrate the operating characteristics of the BAR method in comparison with other methods.
本文研究了一种稀疏线性回归估计量（称为折断自适应岭估计量，简称BAR估计量）的渐近性质，该估计量由一种基于(\ell_1)的迭代加权(\ell_2)惩罚算法产生，以岭估计量作为其初始值。我们证明，BAR估计量在变量选择上是一致的，并且在参数估计上具有神谕性质。此外，我们证明BAR估计量具有分组效应：高度相关的协变量自然地聚集在一起，这是其他神谕变量选择方法所不具备的理想性质。最后，我们将BAR与一种稀疏限制最小二乘估计量相结合，并给出了在高维或超高维设置下，所得的两阶段稀疏回归方法除具有分组性质外，在变量选择和估计上一致的条件。进行了数值研究，以调查和说明BAR方法与其他方法相比的操作特性。
