Robust linear estimation with second order statistics information uncertainty

In this paper, we develop a robust linear estimation (RLE) in presence of a priori statistical information with uncertainties without a model of a with uncertainty but without assumption of model of parameter under estimation and observation. We assume that a random vector x is observed through a nonlinear (or linear) transformation y = f(x, w), where w is noise. We consider the case that there are some uncertainties in second order statistical information of x and y, i.e., Cx, C_yx and C_y and propose an optimal minimax linear estimator that minimizes worst case mean-squared error (MSE) in the region of uncertainty. The minimax estimator can be formulated as a solution to a semidefinite programming problem (SDP). We consider both the Frobenius norm and spectral norm of the uncertainty constraints, leading to the two corresponding robust linear estimators. Finally, Numerical examples are given which illustrates the effectiveness of the proposed estimators.