A new test for the proportionality of two large-dimensional covariance matrices

Abstract

Let X 1 , … , X n 1 + 1 ∼ iid N p ( μ 1 , Σ 1 ) and Y 1 , … , Y n 2 + 1 ∼ iid N p ( μ 2 , Σ 2 ) be two independent random samples, where p < n 2 . In this article, we propose a new test for the proportionality of two large p × p covariance matrices Σ 1 and Σ 2 . By applying modern random matrix theory, we establish the asymptotic normality property for the proposed test statistic as ( p , n 1 , n 2 ) → ∞ together with the ratios p / n 1 → y 1 ∈ ( 0 , ∞ ) and p / n 2 → y 2 ∈ ( 0 , 1 ) under suitable conditions. We further showed that these conclusions are still valid if normal populations are replaced by general populations with finite fourth moments.