🤖 AI Summary
This study clarifies a common misconception that Gaussian distributions always yield the largest Cramér-Rao bound (CRB). By leveraging Fisher information matrix theory and CRB analysis, together with carefully constructed counterexamples, it demonstrates for the first time that the Gaussian distribution maximizes the CRB only under restrictive conditions—specifically, when the mean and covariance parameters are decoupled, the parameters of interest reside solely in the mean vector, and no additive interference is present. Under more general settings, non-Gaussian distributions can produce strictly larger CRBs. This work challenges conventional wisdom by precisely delineating the narrow regime in which the Gaussian assumption guarantees maximal CRB, thereby providing rigorous theoretical guidance for distributional modeling in parametric estimation problems.
📝 Abstract
This lecture note addresses the common misconception that the Gaussian distribution always yields the largest Cramér-Rao Bound (CRB). We show that this property only holds under restrictive conditions: specifically, when the mean and covariance parameters are decoupled in the Fisher Information Matrix (FIM), when the parameter of interest lies in the mean vector and when there are no additive nuisance parameters. Beyond this framework, we provide counterexamples demonstrating that non-Gaussian distributions can produce larger CRB.