CramérRaoGrenze

The Cramér-Rao bound, also known as the Cramér-Rao-Villiers bound or simply the Cramér-Rao limit, is a fundamental concept in estimation theory. It provides a lower bound on the variance of any unbiased estimator for a particular parameter. This means that no unbiased estimator can have a variance smaller than the Cramér-Rao bound. The bound is derived using information theory principles, specifically the Fisher information.

The Fisher information, denoted as $I(\theta)$, quantifies the amount of information that an observable random variable $X$ carries about an unknown parameter $\theta$. A higher Fisher information indicates that the data is more informative about the parameter, leading to a potentially more precise estimation. The Cramér-Rao bound for an unbiased estimator $\hat{\theta}$ of $\theta$ is given by $1/I(\theta)$.

If an unbiased estimator achieves this lower bound, it is called an efficient estimator. Efficient estimators are desirable because they achieve the lowest possible variance among all unbiased estimators for a given parameter. Finding an efficient estimator is often a goal in statistical inference. The Cramér-Rao bound is crucial for assessing the performance of estimators and for understanding the theoretical limits of statistical estimation. It is widely used in various fields, including signal processing, econometrics, and machine learning.