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''hyperefficient 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.
 
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091210/s0912101.png" /> be independent identically-distributed random variables that take values in a sampling space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091210/s0912102.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091210/s0912103.png" />. Suppose that the family of distributions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091210/s0912104.png" /> is such that there is a consistent sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091210/s0912105.png" /> of maximum-likelihood estimators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091210/s0912106.png" /> of the parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091210/s0912107.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091210/s0912108.png" /> be a sequence of asymptotically-normal estimators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091210/s0912109.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091210/s09121010.png" />. If
+
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 ) }
 +
 
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091210/s09121011.png" /></td> </tr></table>
+
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
  
for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091210/s09121012.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091210/s09121013.png" /> is the [[Fisher amount of information|Fisher amount of information]], and if, in addition, the strict inequality
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$$ \tag{* }
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\lim\limits _ {n \rightarrow \infty } \
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{\mathsf E} _ {\theta  ^ {*}  }
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[ n ( T _ {n} - \theta  ^ {*} )  ^ {2} ]
 +
\frac{1}{I ( \theta  ^ {*} ) }
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091210/s09121014.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
+
$$
  
holds at least at one point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091210/s09121015.png" />, then the sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091210/s09121016.png" /> is called superefficient relative to the quadratic loss function, and the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091210/s09121017.png" /> at which (*) holds are called points of superefficiency.
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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====
 
====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>
 
<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>

Latest revision as of 08:24, 6 June 2020


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