CramérRaorajoitetta

CramérRaorajoitetta refers to a fundamental inequality in probability theory and statistics that provides a lower bound for the variance of an unbiased estimator. It is named after Harald Cramér and Rao, who independently derived this result. The inequality establishes that the variance of any unbiased estimator for a parameter cannot be less than a certain value determined by the Fisher information of the data.

The CramérRaorajoitetta is expressed mathematically as Var(T) >= 1 / I(θ), where T is an unbiased estimator of a parameter θ, and I(θ) is the Fisher information of the underlying probability distribution with respect to θ. The Fisher information quantifies the amount of information that an observable random variable X carries about an unknown parameter θ upon which the probability of X depends.

This inequality is significant because it sets a theoretical limit on the precision achievable by unbiased estimators. If an estimator achieves this lower bound, it is termed a "<bos> Cramér-Rao efficient estimator." Such estimators are considered optimal in terms of variance among all unbiased estimators. Finding an estimator that achieves the CramérRaorajoitetta is a key goal in statistical estimation theory. The inequality is widely used to assess the quality of estimators and to guide the development of new estimation methods.