Efficient estimator

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An unbiased statistical estimator whose variance is the lower bound in the Rao–Cramér inequality. An efficient estimator is a sufficient statistic for the parameter to be estimated. If an efficient estimator exists, then it can be obtained by the maximum-likelihood method. Owing to the fact that in many cases the lower bound in the Rao–Cramér inequality cannot be attained, in mathematical statistics an efficient estimator is frequently defined as one having minimal variance in the class of all unbiased estimators (cf. Unbiased estimator) of the parameter in question.


[1] H. Cramér, "Mathematical methods of statistics" , Princeton Univ. Press (1946)
[2] I.A. Ibragimov, R.Z. [R.Z. Khas'minskii] Has'minskii, "Statistical estimation: asymptotic theory" , Springer (1981) (Translated from Russian)
[3] C.R. Rao, "Linear statistical inference and its applications" , Wiley (1965)
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Efficient estimator. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by M.S. Nikulin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article