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 Lr(Pϑ)-differentiability, requiring strong differentiability in Lr(Pϑ) of the rth root of the likelihood ratio relative to Pϑ.