= Efficient unbiased estimator implies exponential family
In a regular one-parameter statistical model, equality in the <Cramér-Rao lower bound> for an unbiased estimator $T$ implies the <score function> $\partial_\theta\log f(x;\theta)=I(\theta)(T(x)-\theta)$. Integrating in the parameter on a regular connected interval gives $f(x;\theta)=h(x)\exp(A(\theta)T(x)-B(\theta))$, with $A'=I$ and $B'=\theta I$. Thus efficiency for every parameter value forces an <exponential family> structure, under the usual common-support and differentiation assumptions.
Back to article page
