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

Let X1,…,Xn1+1∼iidNp(μ1,Σ1) and Y1,…,Yn2+1∼iidNp(μ2,Σ2) be two independent random samples, where p<n2. In this article, we propose a new test for the proportionality of two large p×p covariance matrices Σ1 and Σ2. By applying modern random matrix theory, we establish the asymptotic normality property for the proposed test statistic as (p,n1,n2)→∞ together with the ratios p/n1→y1∈(0,∞) and p/n2→y2∈(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.