CLT for large dimensional general Fisher matrices and its applications in high-dimensional data analysis

Abstract: Random Fisher matrices arise naturally in multivariate statistical analysis and understanding the properties of its eigenvalues is of primary importance for many hypothesis testing problems like testing the equality between two multivariate population covariance matrices, or testing the independence between sub-groups of a multivariate random vector. This paper is concerned with the properties of a large-dimensional Fisher matrix when the dimension of the population is proportionally large compared to the sample size. Most of existing works on Fisher matrices deal with a particular Fisher matrix where populations have i.i.d components so that the population covariance matrices are all identity. In this paper, we consider general Fisher matrices with arbitrary population covariance matrices. The first main result of the paper establishes the limiting distribution of the eigenvalues of a Fisher matrix while in a second main result, we provide a central limit theorem for a wide class of functionals of its eigenvalues. Some applications of these results are also proposed for testing hypotheses on high-dimensional covariance matrices.