# Cramer-Rao bound

(1.1 hours to learn)

## Summary

The Cramer-Rao bound gives the minimum possible variance of an unbiased estimator of the parameters of a probability distribution. It is used to prove the asymptotic efficiency of the maximum likelihood estimator.

## Context

This concept has the prerequisites:

- Fisher information (The bound is given in terms of Fisher information.)
- covariance (The proof of the theorem uses properties of covariance.)
- partial derivatives (The proof of the theorem uses partial derivatives.)

## Goals

- Prove the Cramer-Rao theorem, which bounds the variance of any unbiased estimator of model parameters.

- Use the result to compute the asymptotic relative efficiency of an estimator.

## Core resources (read/watch one of the following)

## -Paid-

→ Probability and Statistics

An introductory textbook on probability theory and statistics.

Location:
Section 8.8, "Fisher information," subsections "The information inequality" and "Efficient estimators," pages 518-522

→ Mathematical Statistics and Data Analysis

An undergraduate statistics textbook.

Location:
Section 8.7, "Efficiency and the Cramer-Rao lower bound," pages 298-305

## See also

- The bound implies that maximum likelihood estimation is asymptotically efficient .