Statistical theory and methods for high-dimensional data

在现代统计分析中，从遗传学和生物学到工程和金融等领域越来越多地遇到高维数据。在遗传学中，微阵列和下一代测序技术可以同时测量单个生物样品中数万个基因的表达水平。在金融领域，人们对数百种金融资产的价格进行高频观察和分析，以进行投资组合配置。然而，高维数使得许多经典的多元统计不合适或不确定。其中一个主要原因是，许多多变量统计涉及样本协方差矩阵，这不是一个很好的估计，其人口对应的高维设置。在未来五至十年内，本研究计划将专注于在以下领域开发新的统计理论和方法：高维均值向量，协方差矩阵和联合分布的检验统计量和估计程序，具有大样本和高维渐近性质。该研究项目的开展将对高维数据的日常统计分析产生重大影响，并促进广泛的科学调查的进展。 这项研究计划的意义是双重的。它将产生新的统计方法和严谨的理论发展，这是大数据时代分析高维数据的迫切需求。它将提供高素质的人才进行理论发展和高维统计的数值研究的宝贵机会。

In modern statistical analysis, high-dimensional data are increasingly encountered in fields ranging from genetics and biology to engineering and finance. In genetics, microarray and next-generation sequencing technology can simultaneously measure expression levels of tens of thousands of genes in a single biological sample. In finance, the prices of hundreds of financial assets are observed at high-frequency and analyzed for portfolio allocation. However, the high dimensionality renders many classical multivariate statistics inappropriate or undefined. One of the primary reasons is that many multivariate statistics involve the sample covariance matrix, which is not a good estimate of its population counterpart in high-dimensional settings. In the next five to ten years, this research program will focus on developing novel statistical theory and methods in the following areas: test statistics and estimation procedures, which enjoy large-sample and high-dimension asymptotic properties, for high-dimensional mean vectors, covariance matrices and joint distributions. The development of this research program will have significant impact on everyday statistical analysis for high-dimensional data, and facilitate advances in a wide range of scientific investigations. The significance of this research program is twofold. It will generate novel statistical methodologies and rigorous theoretical development, which are in urgent demand for analyzing high-dimensional data in the era of big data. It will offer Highly Qualified Personnel valuable opportunities to conduct theoretical development and numerical investigations of high-dimensional statistics.