Statistics Definitions > Cramer-Rao Lower Bound

## What is the Cramer-Rao Lower Bound?

The **Cramer-Rao Lower Bound (CRLB) **gives a lower estimate for the variance of an unbiased estimator. Estimators that are close to the CLRB are more unbiased (i.e. more preferable to use) than estimators further away.

**The Cramer-Rao Lower bound is theoretical;** Sometimes a perfectly unbiased estimator (i.e. one that meets the CRLB) doesn’t exist. Additionally, the CRLB is difficult to calculate unless you have a very simple scenario. Easier, general, alternatives for finding the best estimator do exist. You may want to consider running a more practical alternative for point estimation, like the Method of Moments.

The CLRB can be used for a variety of reasons, including:

**Creating a benchmark**for a best possible measure — against which all other estimators are measured. If you have several estimators to choose from, this can be very useful.**Feasibility studies**to find out if it’s possible to meet specifications (e.g. sensor usefulness).- Can occasionally provide form for MVUE.

## Methods

There are a couple of different ways you can calculate the CRLB. The most common form, which uses Fisher information is:

Let X

_{1}, X_{2},…X_{n}be a random sample with pdf f (x,Θ). If is an unbiased estimator for Θ, then:

Where:

Is the Fisher Information.

You can find examples of hand calculations here.

## Calculating the CLRB with Software

At the time of writing, none of the major software packages (like SPSS, SAS or MAPLE) have built in commands for calculating the Cramer-Rao Lower Bound. This download (an unofficial add-in) is available for MATLAB.

Part of the reason for the lack of software is that the CLRB is distribution specific; In other words, different distributions have different tips and tricks to finding it. The computations are outside the scope of this article, but you can find a couple of examples here (for a binomial distribution) and here (for a normal distribution).

## Other Names

The Cramer-Rao Lower Bound is also called:

- Cramer-Rao Bound (CRB),
- Cramer-Rao inequality,
- Information inequality,
- Rao-Cramér Lower Bound and Efficiency.

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