##  [Cramér–Rao Bound](/index.php/cramer-rao-bound) 

  ##  [Cramér–Rao Bound](https://natural.quantumdictionary.io/cramer-rao-bound-0) 

  

 [![Natural & Formal Sciences Dictionary](/sites/default/files/styles/large/public/2026-01/Natural%20%26%20Formal%20Sciences.png.webp?itok=2kCDRVQv)](/index.php/topic-specific-dictionaries/natural-formal-sciences)



**Natural & Formal Sciences Dictionary**

 







 

 

 

 



 

 

 

 

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).

 

 

 

 

 





 

 



 ##  [Cramér–Rao Bound](https://information.quantumdictionary.io/cramer-rao-bound-1) 

  

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**Information & Communication Dictionary**

 







 

 

 

 



 

 

 

 

Definition

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.