Cramer-Rao Bound for Arbitrarily Constrained Sets

TL;DR

This paper derives a generalized Cramer-Rao bound applicable to any constrained parameter set, unifying various existing bounds and providing a broad framework for understanding estimation accuracy under constraints.

Contribution

It introduces a CRB based on the tangent cone that applies to arbitrary constraints, extending beyond traditional equality, inequality, or manifold constraints.

Findings

The CRB applies to any constrained set and estimation bias.

It unifies and generalizes existing bounds.

The tangent cone governs the impact of constraints on estimation accuracy.

Abstract

This paper presents a Cramer-Rao bound (CRB) for the estimation of parameters confined to an arbitrary set. Unlike existing results that rely on equality or inequality constraints, manifold structures, or the nonsingularity of the Fisher information matrix, the derived CRB applies to any constrained set and holds for any estimation bias and any Fisher information matrix. The key geometric object governing the new CRB is the tangent cone to the constraint set, whose span determines how the constraints affect the estimation accuracy. This CRB subsumes, unifies, and generalizes known special cases, offering an intuitive and broadly applicable framework to characterize the minimum mean-square error of constrained estimators.