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The Cramér–Rao bound is stated in this section for several increasingly general cases, beginning with the case in which the parameter is a scalar and its estimator is unbiased.
Equivalently, the estimator achieves equality in the Cramér–Rao inequality for all θ. The Cramér–Rao lower bound is a lower bound of the variance of an unbiased estimator, representing the "best" an unbiased estimator can be. An efficient estimator is also the minimum variance unbiased estimator (MVUE). This is because an efficient ...
For the Cramér–Rao inequality and the Rao–Blackwell theorem see the relevant entries on Earliest Known Uses of Some of the Words of Mathematics; For Rao contribution to information geometry Cramer-Rao Lower Bound and Information Geometry; Photograph of Rao with Harald Cramér in 1978 C. R. Rao from the PORTRAITS OF STATISTICIANS
A Rao–Blackwell estimator δ 1 (X) of an unobservable quantity θ is the conditional expected value E(δ(X) | T(X)) of some estimator δ(X) given a sufficient statistic T(X). Call δ( X ) the "original estimator" and δ 1 ( X ) the "improved estimator" .
Information inequality may mean in statistics, the Cramér–Rao bound , an inequality for the variance of an estimator based on the information in a sample in information theory, inequalities in information theory describes various inequalities specific to that context.
The quantum Cramér–Rao bound is the quantum analogue of the classical Cramér–Rao bound. It bounds the achievable precision in parameter estimation with a quantum system: It bounds the achievable precision in parameter estimation with a quantum system:
That is, there is a decomposition for which the second inequality is saturated, which is the same as stating that the quantum Fisher information is the convex roof of the variance over four, discussed above. There is also a decomposition for which the first inequality is saturated, which means that the variance is its own concave roof [14]
In statistics a minimum-variance unbiased estimator (MVUE) or uniformly minimum-variance unbiased estimator (UMVUE) is an unbiased estimator that has lower variance than any other unbiased estimator for all possible values of the parameter.