Cramér–Rao bound

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 $\psi (\theta )$ satisfiesIndeed, differentiating ${\mathbb{E}}_{\theta }T=\psi (\theta )$ gives ${\mathrm{Cov}}_{\theta }(T,S_{\theta })={\psi }^{\prime }(\theta )$, and the Cauchy-Schwarz inequality gives the bound.