Rethinking Biased Estimation: Improving Maximum Likelihood and the Cramér–Rao Bound
Foundations and Trends in Signal Processing
Structured least squares problems and robust estimators
IEEE Transactions on Signal Processing
Detection of Gaussian constellations in MIMO systems under imperfect CSI
IEEE Transactions on Communications
Structured Total Maximum Likelihood: An Alternative to Structured Total Least Squares
SIAM Journal on Matrix Analysis and Applications
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In this paper, we consider the problem of estimating an unknown deterministic parameter vector in a linear regression model with random Gaussian uncertainty in the mixing matrix. We prove that the maximum-likelihood (ML) estimator is a (de)regularized least squares estimator and develop three alternative approaches for finding the regularization parameter that maximizes the likelihood. We analyze the performance using the Cramer-Rao bound (CRB) on the mean squared error, and show that the degradation in performance due the uncertainty is not as severe as may be expected. Next, we address the problem again assuming that the variances of the noise and the elements in the model matrix are unknown and derive the associated CRB and ML estimator. We compare our methods to known results on linear regression in the error in variables (EIV) model. We discuss the similarity between these two competing approaches, and provide a thorough comparison that sheds light on their theoretical and practical differences.