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''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|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====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  I.A. Ibragimov,   R.Z. [R.Z. Khas'minskii] Has'minskii,   "Statistical estimation: asymptotic theory" , Springer  (1981)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  L. Schmetterer,   "Introduction to mathematical statistics" , Springer  (1974)  (Translated from German)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  L. le Cam,   "On some asymptotic properties of maximum likelihood estimates and related Bayes estimates"  ''Univ. California Publ. Stat.'' , '''1'''  (1953)  pp. 277–330</TD></TR></table>