A Globally Convergent Algorithm for Linear Minimax Regret Estimation of Deterministic Parameters with Bounded Data Uncertainties

Estimating an unknown deterministic parameter vector with a linear estimator in a noisy linear model is well-known in statistics. This paper studies the open problem in <xref ref-type="bibr" rid="ref1" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">[1]</xref>, i.e., developing the minimax regret estimator in the presence of bounded data uncertainties for more general classes of weighting matrices, where the regret is the difference between the mean-squared error (MSE) attainable without knowledge of the true parameters and the optimal MSE attainable with such knowledge. Firstly, we establish that the minimax regret <xref ref-type="disp-formula" rid="deqnmr1" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(MR)</xref> problem is equivalent to a semidefinite relaxation minimax regret <xref ref-type="disp-formula" rid="deqnsdr1" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(SDR-MR)</xref> problem, utilizing SDR techniques and exploring deeper into the relation between the parameter vector in the regret and a rank-one matrix. This equivalence forms the cornerstone of our analysis. Subsequently, we investigate the existence of a saddle point for the SDR-MR problem’s objective function, and directly obtain the MR estimator if one exists. Otherwise, we proceed to employ the projected subgradient method, with the sublinear rate of convergence, to obtain the MR estimator. Notably, the main challenges stem from some non-convex problems involved in the above process, which are effectively addressed by our algorithm designed for SDP with a single linear fractional. Simulations demonstrate the effectiveness of the MR estimator and validate our theoretical results.