Root Mean Square Error of the Maximum Likelihood Estimator of Signals in Gaussian Noise Using the Directional Curvature of the Signal Manifold
Theagenis J. Abatzoglou · 2023
The Cramer-Rao Bound (CRB) is used to predict the Root Mean Square Error (RMSE) of the Maximum Likelihood Estimator (MLE). This is accurate at high signal-to-noise ratios (SNR) but underestimates the RMSE at lower SNR. We derive the RMSE of the MLE for sample vectors at a distance from the signal manifold which corresponds to moderate SNR. We show this is a generalization of the corresponding CRB which depends on the directional radius of curvature of the signal manifold. We show the MLE is singular at the centers of curvature of the manifold. These results are crucial in explaining the “threshold SNR” that occurs when estimating signal parameters at lower SNR.