The Gaussian data assumption does not always lead to the largest CRB
This clarifies a theoretical limitation in statistical estimation for researchers, but it is incremental as it refines an existing understanding rather than introducing new methods.
The paper tackles the misconception that Gaussian data always maximizes the Cramér-Rao Bound (CRB), showing this only holds under restrictive conditions like decoupled mean and covariance parameters, and provides counterexamples where non-Gaussian distributions yield larger CRB.
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.