Innovated Statistical Inference for Complex and High-Dimensional Data

越来越多的高维数据被收集和分析在不同的研究领域。在实践中，数据科学家在探索和理解复杂和高维数据时面临着重大的分析挑战。高维数据的统计推断在统计学、生物统计学、计量经济学、地球科学、机器学习、信号处理等理论和应用研究中是必不可少的。大数据迅速重塑了统计建模，并彻底改变了统计分析。存在着许多挑战和开放性问题，其解决需要创新的理念和技术。该项目将解决高维假设检验中出现的新挑战。测试高维结构参数在估计和量化不确定性、做出明智的选择以及从大数据中发现知识方面起着至关重要的作用。本计画将发展新的统计方法与理论来研究高维假设检验的三个重要课题：（1）高维协方差矩阵的功效增强检验，（2）高维均值向量的功效增强检验，以及（3）高维数据的非线性统计相依性。研究成果将为解决三个研究主题中的开放问题提供强大的分析工具。这些方法和理论是通用的，它们可以直接扩展到解决其他重要的假设检验的高维数据，如测试在高维spiked模型。该奖项反映了NSF的法定使命，并被认为是值得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的支持。

Increasing amounts of high-dimensional data are being collected and analyzed in a diverse range of research areas. In practice, data scientists face significant analytic challenges when exploring and understanding the complex and high-dimensional data. Statistical inference of high-dimensional data is essential in theoretical and applied research of statistics, biostatistics, econometrics, geoscience, machine learning, signal processing, and many others. Big Data has rapidly reshaped statistical modeling and revolutionized statistical analysis. There exist many challenges and open problems, whose solutions require innovative ideas and techniques. This project will address new challenges arising in high-dimensional hypothesis testing. Testing high-dimensional structural parameters plays a vital role in estimating and quantifying uncertainty, making informed choices, and discovering knowledge from Big Data. In this project, novel statistical methods and theory are developed to study three important topics of high-dimensional hypothesis testing: (1) power enhancement tests for high-dimensional covariance matrices, (2) power enhancement tests for high-dimensional mean vectors, and (3) nonlinear statistical dependence of high-dimensional data. The research outcomes will provide powerful analytic tools for solving open problems in three research topics. The methods and theory are general, and they can be directly extended to address other important hypothesis testings for high-dimensional data such as testing in high-dimensional spiked models. Software packages will be developed to make the research outcomes readily available to other researchers and practitioners.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.