Random Matrices and Related Problems

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Methods for statistical analysis of high dimensional data and large random matrices have receive increased attention in recent years. This project aims to develop new methods for these problems and extend the applicability of existing methods. The research on new methodology for high dimensional data includes both tools for ultra-high dimensional data, and the theoretical study of statistical properties for methods applicable to data with high dimensionality and fixed sample sizes. In this project, the PIs will develop new statistical methods and extend existing methods with novel applications in many fields of statistics, physics, and biology. The efficiency of these methodologies will be demonstrated via simulations and applications to real data sets. In addition, the research results will promote teaching and learning activities. The research results will be disseminated through conference presentations and publications. This project develops new methodologies for tests on the dependence structure for high dimensional data and large random matrices. The PIs will investigate a series of topics that are closely related to high dimensional data, including tests on dependence structure for high dimensional data, tests on high-dimensional mean vectors and high-dimensional covariance matrices, and nonparametric tests for complete independence for high-dimensional data. In addition, the PIs will study the spectral radii of large random matrices such as the limiting distribution for the maximum eigenvalue from principal minors of sample covariance matrices, the largest eigenvalues of Markov matrices, and the limiting distribution for the largest absolute value for the products of truncated unitary matrices. The research in this project will expand the scope of the application of high-dimensional statistical methods and large random matrices.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.