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[6] [7] It is also known as Fréchet-Cramér–Rao or Fréchet-Darmois-Cramér-Rao lower bound. It states that the precision of any unbiased estimator is at most the Fisher information ; or (equivalently) the reciprocal of the Fisher information is a lower bound on its variance .
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 ...
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:
The Cramér–Rao bound [9] [10] states that the inverse of the Fisher information is a lower bound on the variance of any unbiased estimator of θ. Van Trees (1968) and Frieden (2004) provide the following method of deriving the Cramér–Rao bound , a result which describes use of the Fisher information.
Comparing this to the variance of the sample mean (determined previously) shows that the sample mean is equal to the Cramér–Rao lower bound for all values of and . In other words, the sample mean is the (necessarily unique) efficient estimator , and thus also the minimum variance unbiased estimator (MVUE), in addition to being the maximum ...
In statistics, the Chapman–Robbins bound or Hammersley–Chapman–Robbins bound is a lower bound on the variance of estimators of a deterministic parameter. It is a generalization of the Cramér–Rao bound ; compared to the Cramér–Rao bound, it is both tighter and applicable to a wider range of problems.
If k exemplars are chosen (without replacement) from a discrete uniform distribution over the set {1, 2, ..., N} with unknown upper bound N, the MVUE for N is +, where m is the sample maximum. This is a scaled and shifted (so unbiased) transform of the sample maximum, which is a sufficient and complete statistic.
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