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.

翻译：本讲义针对一种常见误解进行了探讨，即高斯分布总是产生最大的克拉美-罗界（CRB）。我们证明，该性质仅在限制性条件下成立：具体而言，当均值和协方差参数在费舍尔信息矩阵（FIM）中解耦时，当感兴趣的参数位于均值向量中，且不存在加性干扰参数时。超出这一框架，我们提供了反例，表明非高斯分布可能产生更大的CRB。