##  [Cramér–Rao Bound](/cramer-rao-bound-0) 

 Definition

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

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

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

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

Consequence

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

 

 

 

 

## Reversal

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

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

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

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.