AI 中文总结
本文针对高维且可能非平稳的重尾时间序列，提出逐元素Huber M估计量与逐元素截断估计量，证明其非渐近性质等理论结果，通过数值实验及月度宏观经济数据变点检测验证方法有效性。
AI 中文摘要
本文研究在重尾性、高维性、一般非线性时间依赖及时间序列可能非平稳条件下的自协方差矩阵估计与推断问题。我们考虑两类尾稳健自协方差矩阵估计方法：逐元素Huber M估计量，以及计算效率更高的逐元素截断估计量。两类估计量均旨在达到矩阵最大范数下的尖锐误差界。针对可能非平稳的过程，我们基于函数依赖下Bernstein型不等式的新变体证明了这些估计量的非渐近性质，该结果或具有独立研究价值。此外，我们证明了逐元素截断自协方差估计量的高维高斯近似结果（作为极限分布），还给出了高斯乘子自助法结果以提升实用性。我们的理论结果为非渐近性，能给出关于样本量、维度、矩及时间依赖强度的显式误差界。数值证据被用于支撑理论结果，最后我们展示了所提方法在月度宏观经济数据变点检测中的优势。
英文摘要
In this paper, we study the autocovariance matrix estimation and inference problems under heavy-tailedness, high-dimensionality, general nonlinear temporal dependence, and potentially nonstationarity of time series. We consider two types of tail-robust autocovariance matrix estimation methods: the element-wise Huber's $M$-estimator and a computationally more efficient element-wise truncated estimator. Both estimators are designed to achieve sharp error bounds in matrix max-norm. The nonasymptotic properties of these estimators are proved based on new variants of Bernstein-type inequalities under functional dependence for the potentially nonstationary processes which may be of independent interest. Moreover, we prove a high-dimensional Gaussian approximation result, as a limiting distribution, for our element-wise truncated autocovariance estimator. A Gaussian multiplier bootstrap result is also given to facilitate the practicality. Our theoretical results are nonasymptotic, which gives explicit error bounds in terms of the sample size, dimensionality, moments, and the strength of temporal dependence. Numerical evidence is provided to support our theoretical results. Finally, we illustrate the benefits of the proposed methodology for detecting change points in monthly macroeconomic data.
