Inference on the Eigenvalues of the Normalized Precision Matrix

Recent developments in the spectral theory of Bayesian Networks has led to a need for a developed theory of estimation and inference on the eigenvalues of the normalized precision matrix, $\Omega$ . In this paper, working under conditions where $n\to \infty$ and $p$ remains fixed, we provide multivariate normal asymptotic distributions of the sample eigenvalues of $\Omega$ under general conditions and under normal populations, a formula for second-order bias correction of these sample eigenvalues, and a Stein-type shrinkage estimator of the eigenvalues. Numerical simulations are performed which demonstrate under what generative conditions each estimation technique is most effective. When the largest eigenvalue of $\Omega$ is small the simulations show that the second order bias-corrected eigenvalue was considerably less biased than the sample eigenvalue, whereas the smallest eigenvalue was estimated with less bias using either the sample eigenvalue or the proposed shrinkage method.