A rigorous proof is presented that global attainment of the Cramer-Rao bound is possible only if the underlying family of distributions is exponential. The proof is placed in the context of Lᵣ(P_ϑ)-differentiability, requiring strong differentiability in Lᵣ(P_ϑ) of the rth root of the likelihood ratio relative to P_ϑ.
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Müller-Funk et al. (1989) studied this question.
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