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.

翻译：本文提出了针对约束于任意集合的参数估计问题的克拉美-罗界（CRB）。与现有依赖等式或不等式约束、流形结构或费舍尔信息矩阵非奇异性条件的研究结果不同，所推导的CRB适用于任意约束集，且对任意估计偏差和任意费舍尔信息矩阵均成立。决定新CRB的关键几何对象是约束集的切锥，其张成空间决定了约束如何影响估计精度。该CRB涵盖、统一并推广了已知的特殊情形，为刻画约束估计器的最小均方误差提供了一个直观且广泛适用的理论框架。