Robust regression has attracted a great amount of attention in the literature recently, particularly for taking asymmetricity into account simultaneously and for high-dimensional analysis. However, the majority of research on the topics falls in frequentist approaches, which are not capable of full probabilistic uncertainty quantification. This paper first proposes a new Huberised-type of asymmetric loss function and its corresponding probability distribution which is shown to have the scale-mixture of normals. Then we introduce a new Bayesian Huberised regularisation for robust regression. A by-product of the research is that a new Bayesian Huberised regularised quantile regression is also derived. We further present their theoretical posterior properties. The robustness and effectiveness of the proposed models are demonstrated in the simulation studies and the real data analysis.

翻译：鲁棒回归近年来在文献中引起了广泛关注，特别是在同时考虑非对称性以及高维分析方面。然而，该主题的大部分研究属于频率学派方法，这些方法无法实现完整的概率不确定性量化。本文首先提出一种新的Huber化型非对称损失函数及其对应的概率分布，并证明其具有正态的尺度混合形式。随后，我们引入一种新的贝叶斯Huber化正则化方法用于鲁棒回归。研究的一个副产品是，我们还推导出了一种新的贝叶斯Huber化正则化分位数回归。此外，我们进一步展示了其理论后验性质。通过模拟研究和实际数据分析，验证了所提出模型的鲁棒性和有效性。