Robust mean-squared error estimation in the presence of model uncertainties

• Published in: IEEE transactions on signal processing Vol. 53; no. 1; pp. 168 - 181
• Main Authors: Eldar, Y.C., Ben-Tal, A., Nemirovski, A.
• Format: Journal Article
• Language: English
• Published: New York, NY IEEE 01.01.2005; Institute of Electrical and Electronics Engineers; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Additive noise; Applied sciences; Data uncertainty; Detection, estimation, filtering, equalization, prediction; Economic forecasting; Error analysis; Errors; Estimation error; Estimators; Exact sciences and technology; Information, signal and communications theory; linear estimation; Mathematical analysis; Mathematical models; mean squared error estimation; minimax estimation; Minimax technique; Minimax techniques; Noise robustness; Parameter estimation; robust estimation; Signal and communications theory; Signal, noise; Studies; Telecommunications and information theory; Uncertainty; Vectors; Vectors (mathematics); Weighting; Performance evaluation; Parameter estimation; Uncertainty; Data uncertainty; robust estimation; Minimax method; mean squared error estimation; Robustness; Linear estimation; Mean square error; minimax estimation
• ISSN: 1053-587X; 1941-0476
• DOI: 10.1109/TSP.2004.838933

We consider the problem of estimating an unknown parameter vector x in a linear model that may be subject to uncertainties, where the vector x is known to satisfy a weighted norm constraint. We first assume that the model is known exactly and seek the linear estimator that minimizes the worst-case mean-squared error (MSE) across all possible values of x. We show that for an arbitrary choice of weighting, the optimal minimax MSE estimator can be formulated as a solution to a semidefinite programming problem (SDP), which can be solved very efficiently. We then develop a closed form expression for the minimax MSE estimator for a broad class of weighting matrices and show that it coincides with the shrunken estimator of Mayer and Willke, with a specific choice of shrinkage factor that explicitly takes the prior information into account. Next, we consider the case in which the model matrix is subject to uncertainties and seek the robust linear estimator that minimizes the worst-case MSE across all possible values of x and all possible values of the model matrix. As we show, the robust minimax MSE estimator can also be formulated as a solution to an SDP. Finally, we demonstrate through several examples that the minimax MSE estimator can significantly increase the performance over the conventional least-squares estimator, and when the model matrix is subject to uncertainties, the robust minimax MSE estimator can lead to a considerable improvement in performance over the minimax MSE estimator.