= Cramér-Rao bound
{c}
{wiki=Cramér–Rao_bound}
= Cramér-Rao lower bound
{c}
{synonym}
For a regular scalar model with <score function> $S_\theta$ and <Fisher information> $I(\theta)=\mathbb E_\theta S_\theta^2$, an estimator $T$ with mean $\psi(\theta)$ satisfies
$$
\operatorname{Var}_\theta(T)
\ge\frac{\psi'(\theta)^2}{I(\theta)}.
$$
Indeed, differentiating $\mathbb E_\theta T=\psi(\theta)$ gives $\operatorname{Cov}_\theta(T,S_\theta)=\psi'(\theta)$, and the <Cauchy-Schwarz inequality> gives the bound.
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