具有重尾和方差异方差的超高维期望回归的抗差估计和收缩
Robust Estimation and Shrinkage in Ultrahigh Dimensional Expectile Regression with Heavy Tails and Variance Heterogeneity
摘要：
高维数据的重尾现象和异质性在各个科学领域中普遍存在，给经典统计方法带来了新的挑战。本文将非对称平方损失和huber型稳健技术相结合，提出了一种适用于高维重尾异质数据的稳健期望回归方法。与经典的huber方法不同，我们在两边引入了两个不同的调节参数来解释可能的不对称性，并允许它们发散以减少由鲁棒近似引起的偏差。在正则化框架中，我们采用了一般折叠凹罚函数，如SCAD或MCP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性质，并弄清楚我们的方法如何发挥其作用，以权衡估计精度对重尾分布。此外，注意到鲁棒非对称损失函数是处处可微的，基于我们的理论研究，我们提出了一个有效的一阶优化算法后，局部线性逼近的非凸问题。在不同分布下的仿真研究表明，我们的方法在系数估计，模型选择和异质性检测令人满意的性能。
英文摘要：
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 trades off the estimation accuracy against the heavy-tailed distribution. Also, noting that the robust asymmetric loss function is everywhere differentiable, 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 demonstrates the satisfactory performances of our method in coefficient estimation, model selection and heterogeneity detection.
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