Efficient Recursive Estimators for a Linear, Time-Varying Gaussian Model with General Constraints

The adaptive estimation of a time-varying parameter vector in a linear Gaussian model is considered where we a priori know that the parameter vector belongs to a known arbitrary subset. We consider a family of efficient recursive estimators for this problem: the recursive constrained maximum likelihood (ML) estimator, the recursive affine minimax, and the recursive minimum mean square error (MMSE) estimator. We show that all three estimators can be substantially simplified by using the recursive weighted least squares (RWLS) algorithm in a first step as the RWLS computes the sufficient statistic for this estimation problem. The recursive constrained ML needs to solve an optimization problem in the second step for the case that the RWLS solution does not fulfill the constraint. In case of affine minimax, we have to solve an optimization problem and to perform an affine transform. The MMSE estimator needs to calculate the mean of a truncated Gaussian density in the second step which is done by Monte Carlo integration. A simple rejection scheme is used to take general constraints for the parameter vector into account. An example shows the superior performance of our proposed estimators in comparison to many other estimators.