Performance of the stochastic MV-PURE estimator with explicit modeling of uncertainty

The stochastic MV-PURE estimator is a linear estimator for stochastic linear model that is highly robust to mismatches in model knowledge and which is specially designed for efficient estimation in noisy and ill-conditioned cases. To date, its properties were analyzed in the theoretical settings of perfect model knowledge and thus could not explain clearly the reason behind its superior performance compared to the Wiener filter observed in simulations in practical cases of imperfect model knowledge. In this paper we derive closed form expressions of the mean-square-error (MSE) of both Wiener filter and the stochastic MV-PURE estimator for the case of perturbed singular values of a model matrix in the linear model considered. These expressions provide in particular conditions under which the stochastic MV-PURE estimator achieves smaller MSE not only than Wiener filter, but also than its full-rank version, the minimum-variance distortionless (MVDR) estimator in such settings. We provide numerical simulations confirming the main theoretical results presented.