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).

Principle

No unbiased estimator can have variance smaller than the reciprocal of the Fisher information computed from the model likelihood; equality holds for efficient estimators under regularity conditions.

Demonstration

For i.i.d. Gaussian samples with known variance σ^2 and mean μ, the Fisher information I(μ)=n/σ^2 yields Var(μ̂)≥σ^2/n, and the sample mean attains this bound, being efficient.

Misapplication

Applying the bound to biased estimators, or to models violating regularity (e.g., non-differentiable likelihoods), or interpreting it as achievable in small samples without checking conditions.

Consequence

Provides a benchmark for estimator performance, guides experiment design by maximizing Fisher information, and identifies when an estimator is asymptotically efficient.

Reversal

When parameterization is changed or biased estimators are allowed, lower variance can be achieved at cost of bias; the bound constrains only unbiased estimators in the given parameterization.

Boundary

Holds under standard regularity assumptions (differentiable likelihood, interchange of differentiation and integration); does not apply directly to biased, constrained, or nonparametric estimators without modification.

Semantic Tension

Sometimes mistaken for a universal lower bound for all estimators; it is specific to unbiased estimators for the chosen parameter and depends on the model and parameterization.

Synthesis

The Cramér–Rao bound sets a model-dependent lower limit on variance for unbiased estimators via Fisher information, serving as a performance benchmark and design guide under regularity.