Probabilistically-constrained Estimation of Random Parameters with Unknown Distribution

The problem of estimating random unknown signal parameters in a noisy linear model is considered. It is assumed that the covariance matrices of the unknown signal parameter and noise vectors are known and that the noise is Gaussian, while the distribution of the random signal parameter vector is unknown. Instead of the traditional minimum mean squared error (MMSE) approach, where the average is taken over both the random signal parameters and noise realizations, we propose a linear estimator that minimizes the MSE which is averaged over the noise only. To make our design pragmatic, the minimization is performed for signal parameter realizations whose probability is sufficiently large, while "discarding" low-probability realizations. It is shown that the obtained linear estimator can be viewed as a generalization of the classical Wiener filter