[Paper Review] CLT for large dimensional general Fisher matrices and its applications in high-dimensional data analysis

Shurong Zheng, Zhidong Bai+1 more|arXiv (Cornell University)|May 8, 2014

Random Matrices and Applications26 references3 citations

TL;DR

This paper establishes a central limit theorem (CLT) for linear spectral statistics of large-dimensional general Fisher matrices with arbitrary population covariance matrices, enabling high-dimensional inference. It derives explicit asymptotic mean and variance formulas for eigenvalue functionals, overcoming limitations of prior work that assumed i.i.d. components or equal covariance matrices.

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.

Motivation & Objective

• Address the lack of asymptotic theory for eigenvalues of general Fisher matrices when dimension p grows proportionally with sample sizes n1 and n2.
• Overcome the limitation of existing CLTs that require i.i.d. components or equal population covariance matrices.
• Provide a central limit theorem with explicit, non-random centering and variance terms for linear spectral statistics of Fisher matrices.
• Enable valid inference in high-dimensional multivariate tests such as equality of covariance matrices and independence testing.
• Extend random matrix theory results to non-Gaussian, non-i.i.d. populations with arbitrary fourth moments.

Proposed method

• Derive the limiting spectral distribution of the Fisher matrix F = B1B2⁻¹ under proportional growth of p, n1, n2.
• Use complex analysis and Stieltjes transforms to characterize the limiting spectral measure of F.
• Establish a CLT for linear spectral statistics W_n = ∑f(λ_i^F) using a deterministic centering term derived from the Stieltjes transform.
• Explicitly compute the asymptotic mean and variance of W_n using integrals involving the limiting spectral distribution and the Stieltjes transform.
• Handle arbitrary fourth moments of population components by avoiding reliance on Gaussian assumptions.
• Prove the CLT under general conditions on the population covariance matrices Σ1 and Σ2, not requiring Σ1 = Σ2.

Experimental results

Research questions

• RQ1 What is the limiting distribution of the eigenvalues of a large-dimensional Fisher matrix when the population covariance matrices are arbitrary?
• RQ2 Can a central limit theorem be established for linear spectral statistics of such Fisher matrices with explicit, non-random centering and variance?
• RQ3 How do the asymptotic mean and variance of eigenvalue functionals depend on the population covariance matrices and the sample dimensions?
• RQ4 Can the CLT be derived without assuming i.i.d. components or equal covariance matrices?
• RQ5 What are the implications of this CLT for high-dimensional hypothesis testing in multivariate analysis?

Key findings

• The paper establishes a CLT for linear spectral statistics of large-dimensional Fisher matrices with arbitrary population covariance matrices, valid under general conditions including non-i.i.d. components.
• The asymptotic mean and variance of the linear spectral statistics are explicitly derived in terms of the Stieltjes transform and the limiting spectral distribution.
• The centering term in the CLT is deterministic and non-random, enabling practical inference, unlike prior results where centering terms were random.
• The asymptotic variance formula involves integrals over the population spectral distribution and the Stieltjes transform, with explicit dependence on the ratio of dimensions.
• The CLT holds even when the fourth moments of the population components are arbitrary, removing a key restriction in earlier work.
• The results provide a theoretical foundation for testing high-dimensional covariance matrices, including equality and independence hypotheses, under general settings.