Cramér–Rao Bound

A lower bound on the variance of any unbiased estimator of a parameter, given by the inverse of the Fisher information: Var(θ̂) ≥ 1 / I(θ).

Under standard regularity conditions for a parametric statistical model with likelihood p(x; θ), the variance of any unbiased estimator (θ̂) of a scalar parameter θ is lower-bounded by the reciprocal of the Fisher information: Var(θ̂) ≥ 1 / I(θ), where I(θ)=E[(∂/∂θ log p(X;θ))^2]. In the multivariate case Cov(θ̂) ≥ I(θ)^{-1} in the matrix sense.