Confidence assessment for spectral estimation based on estimated covariances

In probability theory, time series analysis, and signal processing, many identification and estimation methods rely on covariance estimates as an intermediate statistics. Errors in estimated covariances propagate and degrade the quality of the estimation result. In particular, in large network systems where each system node of the network gather and pass on results, it is important to know the reliability of the information so that informed decisions can be made. In this work, we design confidence regions based on covariance estimates and study how these can be used for spectral estimation. In particular, we consider three different confidence regions based on sets of unitarily invariant matrices and bound the eigenvalue distribution based on three principles: uniform bounds; arithmetic and harmonic means; and the Marcenko-Pastur Law eigenvalue distribution for random matrices. Using these methodologies we robustly bound the energy in a selected frequency band, and compare the resulting spectral bound from the respective confidence regions.