The Gaussian data assumption does not always lead to the largest CRB

TL;DR

This paper clarifies that the Gaussian assumption does not always produce the largest CRB, especially when certain conditions are not met, and provides counterexamples with non-Gaussian distributions.

Contribution

It identifies specific conditions where Gaussian distributions maximize the CRB and presents counterexamples outside those conditions showing non-Gaussian distributions can have larger CRB.

Findings

Gaussian assumption yields largest CRB only under restrictive conditions

Counterexamples show non-Gaussian distributions can have larger CRB

Conditions include decoupled mean and covariance, mean parameter of interest, no nuisance parameters

Abstract

This lecture note addresses the common misconception that the Gaussian distribution always yields the largest Cram\'er-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.