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The Cramér-Rao lower bound, often abbreviated as CRLB, is a fundamental result in statistical inference that provides a theoretical limit on the variance of any unbiased estimator for a parameter. Developed independently by Harald Cramér in 1945 and Rao in 1945, it establishes a minimum possible variance that no unbiased estimator can achieve for estimating a parameter of a probability distribution.

The CRLB is derived from the Fisher information, which quantifies the amount of information a random variable carries about an unknown parameter of a distribution. The Fisher information is the expected value of the square of the score function, where the score function is the derivative of the log-likelihood function with respect to the parameter. The CRLB states that the variance of any unbiased estimator of a parameter is greater than or equal to the reciprocal of the Fisher information.

If an unbiased estimator achieves this lower bound, it is called an efficient estimator. Efficient estimators are desirable because they have the smallest possible variance among all unbiased estimators, meaning they are the most precise. However, efficient estimators do not always exist. The CRLB serves as a benchmark, indicating how close an estimator's variance is to the theoretical minimum. It is a crucial tool for evaluating the performance of estimators and for guiding the search for optimal estimation methods in various fields, including signal processing, communications, and econometrics.