Cramer-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.

An efficient estimator attains the relevant lower bound on estimator variance, such as the Cramer-Rao bound in a regular parametric model.

The van Trees inequality is a Bayesian Cramer--Rao bound. For scalar likelihood information $I(\theta )$ and a differentiable prior density $\pi$ vanishing at its boundary,It follows by integration by parts and the Cauchy--Schwarz inequality applied to the joint likelihood-prior score.