Error bounds in constrained estimation.

Parameter constraints are employed in a variety of situations in multidimensional estimation problems to incorporate prior information, design constraints, or to perform model reduction or approximation. When parameter constraints are introduced, the fundamental structure of the estimation problem is changed, and as a consequence, commonly used lower bounds on estimator error need to be modified to account for the constraints. In this dissertation, we derive general finite-sample Cramer-Rao (CR) and Bhattacharyya-type lower bounds that incorporate smooth parameter constraints. We refer to these bounds as the constrained CR and constrained Bhattacharyya bounds. In contrast to other problem-specific approaches to deriving performance bounds for constrained estimation problems, our approach does not require a global reparameterization of the parameter space. The constrained CR and Bhattacharyya bounds are derived as the local limit of a more general multivariate Chapman-Robbins-type covariance bound and are shown to depend only on local geometric properties of the likelihood function and the parameter constraints. These geometric properties reveal a simple mechanism by which parameter constraints reduce achievable estimator error covariance. In particular, constraints are incorporated into the constrained CR and constrained Bhattacharyya bounds through oblique projections of the associated Fisher information matrices onto local Taylor polynomial approximations to the constraint space. The necessary and sufficient conditions under which the constrained bounds are achievable are shown to be similar to those required for the standard CR and Bhattacharyya bounds. We use the constrained bounds to study the impact of parameter constraints on achievable estimator error covariance in a number of representative multidimensional constrained-parameter estimation problems occurring signal processing. Specific examples considered include: linear constraints for Gaussian linear models, object support constraints in image reconstruction, signal subspace constraints in sensor array processing, and average power constraints in spectral estimation and signal extraction.PhDElectrical Engineering: SystemsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/105536/1/9135603.pdfDescription of 9135603.pdf : Restricted to UM users only