Collaborative Research: Asymptotic Statistical Inference for High-dimensional Time Series

信息时代见证了广泛领域的高维时间序列数据收集的爆炸式增长，包括金融、信号处理、神经科学、气象学、地震学等。对于低维时间序列，已有较完善的估计和推理理论。高维环境下的推理理论具有重要的基础和广泛的应用，但目前研究较少。研究人员在解决现实问题时面临许多挑战：(i)数据生成系统的复杂动力学，（ii）时间和截面依赖性，（iii）高维性和（iv）非高斯分布。本课题的目标是发展和推进高维时间序列数据的推理理论。该项目将向研究生提供培训，并向公众提供统计资料包。本项目涉及发展高维时间序列估计和推理的系统渐近理论，包括参数估计、同时置信区间的构建、预测、模型选择、格兰杰因果检验、假设检验和谱域估计。为此，提出了一种新的高维时间序列参数和二阶特征估计方法。为高维时间序列的渐近分析将发展新的工具和浓度不等式。为了同时进行推理和显著性检验，pi将研究非常深的高斯近似问题和高维中心极限定理，同时考虑高维、时间和截面依赖关系。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

The information era has witnessed an explosion in the collection of high dimensional time series data across a wide range of areas, including finance, signal processing, neuroscience, meteorology, seismology, among others. For low dimensional time series, there is a well-developed estimation and inference theory. Inference theory in the high dimensional setting is of fundamental importance and has wide applications, but has been rarely studied. Researchers face a number of challenges in solving real-world problems: (i) complex dynamics of data generating systems, (ii) temporal and cross-sectional dependencies, (iii) high dimensionality and (iv) non-Gaussian distributions. The goal of this project is to develop and advance inference theory for high dimensional time series data by concerning all the above characteristics. The project will provide training to graduate students and publicly avaialble statistical packages. This project involves developing a systematic asymptotic theory for estimation and inference for high dimensional time series, including parameter estimation, construction of simultaneous confidence intervals, prediction, model selection, Granger causality test, hypothesis testing, and spectral domain estimation. To this end, a new methodology for the estimation of parameters and second-order characteristics for high dimensional time series will be proposed. New tools and concentration inequalities for the asymptotic analysis of high-dimensional time series will be developed. To perform simultaneous inference and significance testing, the PIs will investigate the very deep Gaussian approximation problem and the high dimensional central limit theorems by taking both high dimensionality and temporal and cross-sectional dependencies into account.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.