On the Attainment of the Cramer-Rao Bound in $\mathbb{L}_r$-Differentiable Families of Distributions

Abstract

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 $\mathbb{L}_r(P_\vartheta)$-differentiability, requiring strong differentiability in $\mathbb{L}_r(P_\vartheta)$ of the $r$th root of the likelihood ratio relative to $P_\vartheta$.