Estimation of two high-dimensional covariance matrices and the spectrum of their ratio

Let Sp,1, Sp,2 be two independent p×p sample covariance matrices with degrees of freedom n1 and n2, respectively, whose corresponding population covariance matrices are Σp,1 and Σp,2, respectively. Knowing Sp,1, Sp,2, this article proposes a class of estimators for the spectrum (eigenvalues) of the matrix Σp,2Σp,1−1 as well as the pair of the whole matrices (Σp,1,Σp,2). The estimators are created based on Random Matrix Theory. Under mild conditions, our estimator for the spectrum of Σp,2Σp,1−1 is shown to be weakly consistent and the estimator for (Σp,1,Σp,2) is shown to be optimal in the sense of minimizing the asymptotic loss within the class of equivariant estimators as n1,n2,p→∞ with p∕n1→c1∈(0,1), p∕n2→c2∈(0,1)∪(1,∞). Also, our estimators are easy to implement. Even when p is 1000, our estimators can be computed in seconds using a personallaptop.