CramérRao

Cramérrao is commonly used as a shorthand for the Cramér–Rao bound, a fundamental result in estimation theory. The bound provides a theoretical lower limit on the variance of unbiased estimators for a parameter within a statistical model, under regularity conditions on the likelihood function.

In a model with independent observations X1, X2, ..., Xn having density or mass function f(x; θ), where θ is the parameter to be estimated, the Fisher information for a single observation is I(θ) = E[(∂/∂θ log f(X; θ))^2], and for n independent observations I_n(θ) = n I(θ). The Cramér–Rao bound states that for any unbiased estimator θ̂ of θ based on these n observations, Var(θ̂) ≥ 1/I_n(θ) = 1/[n I(θ)]. For a vector parameter θ ∈ R^k, the bound generalizes to Cov(θ̂) ≥ I(θ)^{-1}, where I(θ) is the Fisher information matrix and the inequality is in the sense of positive semidefinite matrices.

Attainability of the bound depends on the model. Equality is possible for efficient estimators in certain regular models; the maximum likelihood estimator is asymptotically efficient, meaning it attains the bound in the limit as sample size grows. In finite samples, the bound may be unattainable, and the bound can be used to assess the efficiency of estimators relative to the best possible unbiased estimator.

Applications of the Cramér–Rao bound include evaluating estimator performance, guiding the design of experiments, and providing a benchmark for comparing estimation methods. The concept remains a core tool in statistics, signal processing, and related fields.