Cramer-Rao Bound for Arbitrarily Constrained Sets

2026-01-27

AI Summary

This work addresses the limitation of existing Cramér-Rao bounds in handling parameter constraints of arbitrary form. It proposes a universal generalized Cramér-Rao bound for parameter estimation under general constraint sets, without requiring assumptions such as equality or inequality constraints, manifold structure, or nonsingularity of the Fisher information matrix. The approach leverages the tangent cone as a central geometric construct, integrating tools from convex analysis and differential geometry to unify and extend prior formulations. The resulting lower bound applies to arbitrary constraint sets, arbitrary estimator biases, and any Fisher information matrix—singular or not—thereby providing an intuitive and broadly applicable benchmark for the minimum mean squared error in constrained parameter estimation.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Reasoning under Uncertainty: Stochastic Optimization
• Intelligent Robots: State Estimation

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Applications of cryptography
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

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.

Problem

• Cramer-Rao bound
• constrained estimation
• arbitrary constraint set
• tangent cone
• Fisher information matrix

Innovation

• Cramer-Rao bound
• arbitrary constraints
• tangent cone
• Fisher information matrix
• constrained estimation