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性增长，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，有一套成熟的估计和推断理论。高维环境下的推理理论具有基础性的重要性和广泛的应用，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力学，(Ii)时间和横截面相关性，(Iii)高维和(Iv)非高斯分布。本项目的目标是通过综合考虑上述特点，发展和完善高维时间序列数据的推理理论。该项目将为研究生提供培训，并向公众提供统计资料包。该项目涉及发展高维时间序列估计和推断的系统渐近理论，包括参数估计、同时可信区间的构造、预测、模型选择、格兰杰因果检验、假设检验和谱域估计。为此，将提出一种新的高维时间序列参数和二阶特征估计方法。将开发用于高维时间序列渐近分析的新工具和浓度不等式。为了进行同时推理和显著性测试，PI将调查非常深的高斯近似问题和高维中心极限定理，同时考虑到高维以及时间和横截面的依赖。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.