# Superefficient estimator

hyperefficient estimator

An abbreviation of the phrase "superefficient sequence of estimators" , used for a consistent sequence of asymptotically-normal estimators of an unknown parameter that is better (more efficient) than a consistent sequence of maximum-likelihood estimators.

Let $X _ {1} \dots X _ {n}$ be independent identically-distributed random variables that take values in a sampling space $( \mathfrak X , {\mathcal B} , {\mathsf P} _ \theta )$, $\theta \in \Theta$. Suppose that the family of distributions $\{ {\mathsf P} _ \theta \}$ is such that there is a consistent sequence $\{ \widehat \theta _ {n} \}$ of maximum-likelihood estimators $\widehat \theta _ {n} = \widehat \theta _ {n} ( X _ {1} \dots X _ {n} )$ of the parameter $\theta$. Let $\{ T _ {n} \}$ be a sequence of asymptotically-normal estimators $T _ {n} = T _ {n} ( X _ {1} \dots X _ {n} )$ of $\theta$. If

$$\lim\limits _ {n \rightarrow \infty } \ {\mathsf E} _ \theta [ n ( T _ {n} - \theta ) ^ {2} ] \leq \frac{1}{I ( \theta ) }$$

for all $\theta \in \Theta$, where $I ( \theta )$ is the Fisher amount of information, and if, in addition, the strict inequality

$$\tag{* } \lim\limits _ {n \rightarrow \infty } \ {\mathsf E} _ {\theta ^ {*} } [ n ( T _ {n} - \theta ^ {*} ) ^ {2} ] < \frac{1}{I ( \theta ^ {*} ) }$$

holds at least at one point $\theta ^ {*} \in \Theta$, then the sequence $\{ T _ {n} \}$ is called superefficient relative to the quadratic loss function, and the points $\theta ^ {*}$ at which (*) holds are called points of superefficiency.

#### References

 [1] I.A. Ibragimov, R.Z. [R.Z. Khas'minskii] Has'minskii, "Statistical estimation: asymptotic theory" , Springer (1981) (Translated from Russian) [2] L. Schmetterer, "Introduction to mathematical statistics" , Springer (1974) (Translated from German) [3] L. le Cam, "On some asymptotic properties of maximum likelihood estimates and related Bayes estimates" Univ. California Publ. Stat. , 1 (1953) pp. 277–330
How to Cite This Entry:
Superefficient estimator. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Superefficient_estimator&oldid=48911
This article was adapted from an original article by M.S. Nikulin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article