🤖 AI Summary
To address the challenge of accurately estimating eigenvalues of precision matrices in high-dimensional settings—where direct inversion of the sample covariance matrix is ill-posed—the paper proposes a novel, inversion-free estimator. Leveraging random matrix theory, it rigorously analyzes the convergence rates of the Stieltjes transform and its derivative of the sample covariance matrix, enabling the construction of a refined eigenvalue estimator. The estimator achieves an asymptotic bias of order $O(1/K^2)$, markedly improving upon the conventional $O(1/K)$ rate; moreover, a rigorous central limit theorem is established, precisely characterizing the asymptotic distribution of the estimator. Theoretical analysis confirms consistency under the high-dimensional asymptotic regime ($n, p o infty$, $p/n o c > 0$), while numerical experiments validate both the enhanced estimation accuracy and the fidelity of the limiting distribution. This work provides the first direct estimation framework for spectral inference of high-dimensional precision matrices that simultaneously achieves higher-order bias correction and verifiable asymptotic normality.
📝 Abstract
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.