 Solution

Assume a common support, a differentiable likelihood, and dominated differentiation under the integrals. The score function $S_{\theta }={\partial }_{\theta }\log f(X;\theta )$ has mean zero. Its second moment is the Fisher information $I(\theta )$. If $T$ is unbiased, differentiation of ${\mathbb{E}}_{\theta }T=\theta$ gives ${\mathbb{E}}_{\theta }(T{S}_{\theta })=1$, hence ${\mathrm{Cov}}_{\theta }(T,S_{\theta })=1$. The Cauchy-Schwarz inequality yieldsthe Cramér-Rao lower bound, assuming positive finite information.

Equality holds precisely when the centred estimator and score are linearly dependent almost surely. The covariance fixes the factor, giving $S_{\theta }=I(\theta )(T-\theta )$. If the same unbiased estimator is efficient for every parameter on a regular interval, integrate this identity in $\theta$:Exponentiation gives $f(x;\theta )=h(x)\exp (A(\theta )T(x)-B(\theta ))$, an exponential family. This proves efficient unbiased estimator implies exponential family under the stated common-support regularity, rather than asserting it for nonregular support-dependent models.