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, . In this paper, working under conditions where and remains fixed, we provide multivariate normal asymptotic distributions of the sample eigenvalues of 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 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.