CramérRaolimiet

CramérRaolimiet is a hypothetical statistical estimation framework that blends elements of the Cramér-Rao lower bound with the Rao-Blackwell theorem, intended to illustrate how nuisance parameters can be incorporated into variance bounds.

Conceptually, it generalizes the classical bound by projecting the score onto a sufficient statistic. Let X have a parametric model Pθ, with score s(X; θ) = ∂ log f(X; θ)/∂θ, and let S(X) be a complete sufficient statistic for θ. Define the Rao-Blackwellized score s̃(X; θ) = Eθ[ s(X; θ) | S(X) ]. The generalized information matrix is Ĩ(θ) = Eθ[ s̃(X; θ) s̃(X; θ)^T ]. The bound states that for any unbiased estimator T(X) of g(θ),

Varθ(T) ≥ [Ĩ(θ)]^{-1}, with equality under suitable regularity when T is a function of S(X) and efficient with respect to Ĩ.

Relation to classical bounds: If S provides no reduction or if T depends on the full data, Ĩ(θ) collapses to the standard Fisher information I(θ), recovering the Cramér-Rao bound. In models with nuisance parameters, the CramérRaolimiet bound can be tighter than the unadjusted bound.

Uses and limitations: The framework is primarily a theoretical tool in teaching and analysis, illustrating how nuisance information can be incorporated. Practical application depends on identifying complete sufficient statistics and verifying regularity conditions.

History: The concept is fictional and used here as an illustrative extension of estimation theory for educational purposes.

See also: Cramér-Rao bound; Rao-Blackwell theorem; Fisher information.