Gaussian data yields the largest CRB only under decoupled mean-covariance in the FIM, mean-vector parameters, and no nuisance parameters; otherwise non-Gaussian distributions can produce larger CRB.
Delmas, Chapter ”Background on real and complex elliptically symmetric distributions”, in Elliptical Distributions in Signal Processing and Machine Learning, Springer 2024
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The Gaussian data assumption does not always lead to the largest CRB
Gaussian data yields the largest CRB only under decoupled mean-covariance in the FIM, mean-vector parameters, and no nuisance parameters; otherwise non-Gaussian distributions can produce larger CRB.