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