Direct Estimation of Eigenvalues of Large Dimensional Precision Matrix

In this paper, we consider directly estimating the eigenvalues of precision matrix, without inverting the corresponding estimator for the eigenvalues of covariance matrix. We focus on a general asymptotic regime, i.e., the large dimensional regime, where both the dimension $N$ and the sample size $K$ tend to infinity whereas their quotient $N/K$ converges to a positive constant. By utilizing tools from random matrix theory, we construct an improved estimator for eigenvalues of precision matrix. We prove the consistency of the new estimator under large dimensional regime. In order to obtain the asymptotic bias term of the proposed estimator, we provide a theoretical result that characterizes the convergence rate of the expected Stieltjes transform (with its derivative) of the spectra of the sample covariance matrix. Using this result, we prove that the asymptotic bias term of the proposed estimator is of order $O(1/K^2)$. Additionally, we establish a central limiting theorem (CLT) to describe the fluctuations of the new estimator. Finally, some numerical examples are presented to validate the excellent performance of the new estimator and to verify the accuracy of the CLT.