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

信息时代已经见证了在广泛的领域，包括金融，信号处理，神经科学，气象学，地震学等收集高维时间序列数据的爆炸。对于低维时间序列，已有成熟的估计和推断理论。高维环境下的推理理论具有重要的理论意义和广泛的应用前景，但研究较少。研究人员在解决现实世界的问题时面临着许多挑战：（i）数据生成系统的复杂动态，（ii）时间和横截面依赖性，（iii）高维性和（iv）非高斯分布。本计画的目标是发展及提升高维时间序列资料之推论理论，以涵盖上述所有特性。该项目将向研究生提供培训，并向公众提供统计数据包。本计画系发展一套系统的高阶时间序列估计与推论的渐近理论，包括参数估计、同时置信区间的建构、预测、模型选择、格兰杰因果关系检验、假设检验、以及谱域估计。为此，将提出一种新的方法估计参数和二阶特征的高维时间序列。新的工具和浓度不等式的高维时间序列的渐近分析将开发。为了同时进行推理和显著性检验，PI将通过考虑高维性以及时间和横截面依赖性来研究非常深的高斯近似问题和高维中心极限定理。该奖项反映了NSF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而被认为值得支持。

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.