Statistical Inference for High-Dimensional Time Series

由于信息技术的快速发展及其在许多科学领域的应用，高维时间序列(HDTS)被常规地收集。对于低维和固定维时间序列的推断方法和理论，当维长相当于或超过时间序列长度时，可能不适用，迫切需要发展新的统计方法，既能适应高维，又能适应时间相关性。HDTS的统计推断是非常重要的，在气候科学、医学成像和金融等学科中有着广泛的应用。本项目旨在开发创新的理论和方法来解决HDTS分析中的几个重要推理问题。该研究建立在自归一化方法的基础上，该方法在处理低维问题上取得了巨大的成功。它向高维语境的推进在方法论和理论上都具有挑战性，它需要新的方法论表述和新的理论。该项目涵盖了HDTS的均值、协方差矩阵和自协方差矩阵的推断，所开发的测试可用于检测变化点、协方差矩阵的特定结构和目标密集备选。在理论方面，将调查基于序列U-统计量的过程的弱收敛，并具有独立的兴趣。该奖项反映了NSF的法定使命，并已被认为值得支持，通过使用基金会的智力价值和更广泛的影响审查标准进行评估。

Due to the rapid development of information technologies and their applications in many scientific fields, high dimensional time series (HDTS) are routinely collected nowadays. The methods and theory developed for the inference of low and fixed dimensional time series may not be applicable when the dimension is comparable to or exceeds time series length, and there is an urgent need to develop new statistical methods that can accommodate both high dimensionality and temporal dependence. Statistical inference for HDTS is fundamentally important and has many applications in disciplines ranging from climate science to medical imaging and finance, among others.This project aims to develop innovative theory and methodologies to address several important inference problems in the analysis of HDTS. The research is built on the self-normalized approach, which has found great success in dealing with low dimensional problems. Its advance to the high dimensional context is challenging both methodologically and theoretically, and it requires a new methodological formulation and new theory. This project covers the inference of the mean, covariance matrix, and auto-covariance matrix for HDTS, and the tests developed can be used to detect change points, certain structure of the covariance matrix and target dense alternative. On the theoretical front, the weak convergence of sequential U-statistic based process will be investigated and is of independent interest.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.