Fan Jianqing, Guo Yongyi, Jiang Bai
Department of Operations Research and Financial Engineering, Princeton University, 98 Charlton Street, Princeton, NJ 08540.
Stoch Process Their Appl. 2022 Aug;150:802-818. doi: 10.1016/j.spa.2019.09.004. Epub 2019 Sep 25.
High-dimensional linear regression has been intensively studied in the community of statistics in the last two decades. For the convenience of theoretical analyses, classical methods usually assume independent observations and sub-Gaussian-tailed errors. However, neither of them hold in many real high-dimensional time-series data. Recently [Sun, Zhou, Fan, 2019, J. Amer. Stat. Assoc., in press] proposed Adaptive Huber Regression (AHR) to address the issue of heavy-tailed errors. They discover that the robustification parameter of the Huber loss should adapt to the sample size, the dimensionality, and the moments of the heavy-tailed errors. We progress in a vertical direction and justify AHR on dependent observations. Specifically, we consider an important dependence structure - Markov dependence. Our results show that the Markov dependence impacts on the adaption of the robustification parameter and the estimation of regression coefficients in the way that the sample size should be discounted by a factor depending on the spectral gap of the underlying Markov chain.
在过去二十年中，高维线性回归在统计学领域得到了深入研究。为便于进行理论分析，经典方法通常假设观测值相互独立且误差服从次高斯尾分布。然而，在许多实际的高维时间序列数据中，这两个假设都不成立。最近[Sun, Zhou, Fan, 2019, 《美国统计协会杂志》，即将发表]提出了自适应Huber回归（AHR）来解决重尾误差问题。他们发现，Huber损失的稳健化参数应适应样本量、维度以及重尾误差的矩。我们在垂直方向上取得了进展，并在相依观测值的情况下证明了AHR的合理性。具体而言，我们考虑一种重要的相依结构——马尔可夫相依。我们的结果表明，马尔可夫相依以这样一种方式影响稳健化参数的自适应以及回归系数的估计，即样本量应根据基础马尔可夫链的谱隙按一个因子进行折扣。
