In this paper, we discuss a family of robust, high-dimensional regression models for quantile and composite quantile regression, both with and without an adaptive lasso penalty for variable selection. We reformulate these quantile regression problems and obtain estimators by applying the alternating direction method of multipliers (ADMM), majorize-minimization (MM), and coordinate descent (CD) algorithms. Our new approaches address the lack of publicly available methods for (composite) quantile regression, especially for high-dimensional data, both with and without regularization. Through simulation studies, we demonstrate the need for different algorithms applicable to a variety of data settings, which we implement in the cqrReg package for R. For comparison, we also introduce the widely used interior point (IP) formulation and test our methods against the IP algorithms in the existing quantreg package. Our simulation studies show that each of our methods, particularly MM and CD, excel in different settings such as with large or high-dimensional data sets, respectively, and outperform the methods currently implemented in quantreg. The ADMM approach offers specific promise for future developments in its amenability to parallelization and scalability.

在本文中，我们讨论了适用于分位数和复合分位数回归的健壮，高维回归模型系列，无论是否针对变量选择使用自适应套索罚分。我们通过应用乘数的交替方向方法（ADMM），主化最小化（MM）和坐标下降（CD）算法来重新构造这些分位数回归问题并获得估计量。我们的新方法解决了（复合）分位数回归（特别是对于高维数据）在有无正则化的情况下缺乏公开可用的方法。通过仿真研究，我们证明了需要适用于各种数据设置的不同算法，这些算法在cqrReg中实现为了进行比较，我们还介绍了广泛使用的内部点（IP）公式，并针对现有的quantreg软件包中的IP算法测试了我们的方法。我们的仿真研究表明，我们的每种方法（尤其是MM和CD）在不同的设置（例如分别使用大型或高维数据集）方面均表现出色，并且优于目前在Quantreg中实施的方法。ADMM方法对并行化和可伸缩性的适应性为未来的发展提供了特殊的希望。