Cramer-Rao Lower Bound (CRLB) Simple Introduction

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.

Cramer-Rao下界（CRLB）给出了无偏估计的方差的较低估计。 接近CLRB的估计比不远处的估计更加无偏（即更优选使用）。

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.

Cramer-Rao下界是理论上的; 有时，不存在完全无偏估计（即符合CRLB的估计）。 此外，除非您有一个非常简单的场景，否则CRLB很难计算。 找到最佳估算器的更简单，通用的替代方案确实存在。 您可能需要考虑为点估计运行更实用的替代方法，例如矩量法。

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.

可以出于各种原因使用CLRB，包括：

• 为最佳可能度量创建基准 - 衡量所有其他估算器的基准。 如果您有多个估算器可供选择，这可能非常有用。
• 可行性研究，以确定是否可以满足规范（例如传感器的有用性）。

Methods

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

您可以通过几种不同的方式计算CRLB。 使用Fisher信息的最常见形式是：

Let X1, X2,…Xn be a random sample with pdf f (x,Θ). If  is an unbiased estimator for Θ, then:



Where:



Is the Fisher Information.

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.

https://www.statisticshowto.datasciencecentral.com/cramer-rao-lower-bound/

文章来源: reborn.blog.csdn.net，作者：李锐博恩，版权归原作者所有，如需转载，请联系作者。

原文链接：reborn.blog.csdn.net/article/details/83861973