CramérRaoVilliers

The Cramér-Rao Lower Bound, often abbreviated as CRLB, is a fundamental concept in statistical estimation theory. It establishes a lower bound on the variance of any unbiased estimator for a parameter. This means that no estimator, no matter how cleverly designed, can achieve a variance smaller than the CRLB. The bound is named after Harald Cramér, C.R. Rao, and Wilhelm Vestergaard Villiers, who independently developed aspects of this theory.

The CRLB is a function of the Fisher information of the parameter being estimated. The Fisher information quantifies how much information a random variable carries about an unknown parameter. A higher Fisher information generally implies a lower CRLB, meaning a more precise estimate is possible.

The significance of the CRLB lies in its ability to assess the performance of estimators. If an estimator achieves the CRLB, it is considered an efficient estimator. However, many practical estimators do not reach the CRLB. In such cases, the CRLB serves as a benchmark, indicating the best possible performance one can hope for. Understanding the CRLB is crucial for researchers and engineers when designing and evaluating statistical models and estimation procedures across various fields, including signal processing, econometrics, and machine learning.