High-dimensional data subject to heavy-tailed phenomena and heterogeneity are commonly encountered in various scientific fields and bring new challenges to the classical statistical methods. In this paper, we combine the asymmetric square loss and huber-type robust technique to develop the robust expectile regression for ultrahigh dimensional heavy-tailed heterogeneous data. Different from the classical huber method, we introduce two different tuning parameters on both sides to account for possibly asymmetry and allow them to diverge to reduce bias induced by the robust approximation. In the regularized framework, we adopt the generally folded concave penalty function like the SCAD or MCP penalty for the seek of bias reduction. We investigate the finite sample property of the corresponding estimator and figure out how our method plays its role to trade off the estimation accuracy against the heavy-tailed distribution. Also, based on our theoretical study, we propose an efficient first-order optimization algorithm after locally linear approximation of the non-convex problem. Simulation studies under various distributions and a real data example demonstrate the satisfactory performances of our method in coefficient estimation, model selection and heterogeneity detection.

遭受重尾现象和异质性影响的高维数据在各个科学领域中都屡见不鲜，这给经典的统计方法带来了新的挑战。在本文中，我们结合了非对称方损失和Huber型鲁棒技术，为超高维重尾异类数据开发了鲁棒的期望回归。与经典的Huber方法不同，我们在两侧引入了两个不同的调整参数以解决可能的不对称性，并允许它们发散以减小由鲁棒逼近引起的偏差。在正则化框架中，我们采用像SCAD或MCP惩罚这样的通常折叠的凹惩罚函数来寻求偏差减少。我们研究了相应估计量的有限样本属性，并指出了我们的方法如何发挥作用来权衡估计精度与重尾分布。同样，基于我们的理论研究，在提出非凸问题的局部线性近似之后，我们提出了一种有效的一阶优化算法。在各种分布下的仿真研究和一个真实的数据示例证明了我们的方法在系数估计，模型选择和异质性检测方面的令人满意的性能。