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A [[Statistical estimator|statistical estimator]] whose values are points in the set of values of the quantity to be estimated.
 
A [[Statistical estimator|statistical estimator]] whose values are points in the set of values of the quantity to be estimated.
  
Suppose that in the realization <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073160/p0731601.png" /> of the random vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073160/p0731602.png" />, taking values in a sample space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073160/p0731603.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073160/p0731604.png" />, the unknown parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073160/p0731605.png" /> (or some function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073160/p0731606.png" />) is to be estimated. Then any statistic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073160/p0731607.png" /> producing a mapping of the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073160/p0731608.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073160/p0731609.png" /> (or into the set of values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073160/p07316010.png" />) is called a point estimator of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073160/p07316011.png" /> (or of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073160/p07316012.png" /> to be estimated). Important characteristics of a point estimator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073160/p07316013.png" /> are its mathematical expectation
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Suppose that in the realization $  x = ( x _ {1} \dots x _ {n} )  ^ {T} $
 +
of the random vector $  X = ( X _ {1} \dots X _ {n} )  ^ {T} $,  
 +
taking values in a sample space $  ( \mathfrak X , {\mathcal B} , {\mathsf P} _  \theta  ) $,  
 +
$  \theta = ( \theta _ {1} \dots \theta _ {k} )  ^ {T} \in \Theta \subset  \mathbf R  ^ {k} $,  
 +
the unknown parameter $  \theta $(
 +
or some function $  g ( \theta ) $)  
 +
is to be estimated. Then any statistic $  T _ {n} = T _ {n} ( X) $
 +
producing a mapping of the set $  \mathfrak X $
 +
into $  \Theta $(
 +
or into the set of values of $  g ( \theta ) $)  
 +
is called a point estimator of $  \theta $(
 +
or of the function $  g ( \theta ) $
 +
to be estimated). Important characteristics of a point estimator $  T _ {n} $
 +
are its mathematical expectation
  
<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/p/p073/p073160/p07316014.png" /></td> </tr></table>
+
$$
 +
{\mathsf E} _  \theta  \{ T _ {n} \}  = \
 +
\int\limits _ {\mathfrak X } T _ {n} ( x)  d {\mathsf P} _  \theta  ( x)
 +
$$
  
 
and the covariance matrix
 
and the covariance matrix
  
<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/p/p073/p073160/p07316015.png" /></td> </tr></table>
+
$$
 +
{\mathsf E} _  \theta  \{ ( T _ {n} - {\mathsf E} \{ T _ {n} \} )
 +
( T _ {n} - {\mathsf E} \{ T _ {n} \} )  ^ {T} \} .
 +
$$
  
The vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073160/p07316016.png" /> is called the error vector of the point estimator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073160/p07316017.png" />. If
+
The vector $  d ( X) = T _ {n} ( X) - g ( \theta ) $
 +
is called the error vector of the point estimator $  T _ {n} $.  
 +
If
  
<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/p/p073/p073160/p07316018.png" /></td> </tr></table>
+
$$
 +
b ( \theta )  = \
 +
{\mathsf E} _  \theta  \{ d ( X) \}  = \
 +
{\mathsf E} _  \theta  \{ T _ {n} \} - g ( \theta )
 +
$$
  
is the zero vector for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073160/p07316019.png" />, then one says that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073160/p07316020.png" /> is an unbiased estimator of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073160/p07316021.png" /> or that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073160/p07316022.png" /> is free of systematic errors; otherwise, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073160/p07316023.png" /> is said to be biased, and the vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073160/p07316024.png" /> is called the bias or systematic error of the point estimator. The quality of a point estimator can be defined by means of the risk function (cf. [[Risk of a statistical procedure|Risk of a statistical procedure]]).
+
is the zero vector for all $  \theta \in \Theta $,  
 +
then one says that $  T _ {n} $
 +
is an unbiased estimator of $  g ( \theta ) $
 +
or that $  T _ {n} $
 +
is free of systematic errors; otherwise, $  T _ {n} $
 +
is said to be biased, and the vector $  b ( \theta ) $
 +
is called the bias or systematic error of the point estimator. The quality of a point estimator can be defined by means of the risk function (cf. [[Risk of a statistical procedure|Risk of a statistical procedure]]).
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H. Cramér,  "Mathematical methods of statistics" , Princeton Univ. Press  (1946)</TD></TR><TR><TD valign="top">[2]</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></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H. Cramér,  "Mathematical methods of statistics" , Princeton Univ. Press  (1946)</TD></TR><TR><TD valign="top">[2]</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></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E.L. Lehmann,  "Theory of point estimation" , Wiley  (1983)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E.L. Lehmann,  "Theory of point estimation" , Wiley  (1983)</TD></TR></table>

Latest revision as of 08:06, 6 June 2020


A statistical estimator whose values are points in the set of values of the quantity to be estimated.

Suppose that in the realization $ x = ( x _ {1} \dots x _ {n} ) ^ {T} $ of the random vector $ X = ( X _ {1} \dots X _ {n} ) ^ {T} $, taking values in a sample space $ ( \mathfrak X , {\mathcal B} , {\mathsf P} _ \theta ) $, $ \theta = ( \theta _ {1} \dots \theta _ {k} ) ^ {T} \in \Theta \subset \mathbf R ^ {k} $, the unknown parameter $ \theta $( or some function $ g ( \theta ) $) is to be estimated. Then any statistic $ T _ {n} = T _ {n} ( X) $ producing a mapping of the set $ \mathfrak X $ into $ \Theta $( or into the set of values of $ g ( \theta ) $) is called a point estimator of $ \theta $( or of the function $ g ( \theta ) $ to be estimated). Important characteristics of a point estimator $ T _ {n} $ are its mathematical expectation

$$ {\mathsf E} _ \theta \{ T _ {n} \} = \ \int\limits _ {\mathfrak X } T _ {n} ( x) d {\mathsf P} _ \theta ( x) $$

and the covariance matrix

$$ {\mathsf E} _ \theta \{ ( T _ {n} - {\mathsf E} \{ T _ {n} \} ) ( T _ {n} - {\mathsf E} \{ T _ {n} \} ) ^ {T} \} . $$

The vector $ d ( X) = T _ {n} ( X) - g ( \theta ) $ is called the error vector of the point estimator $ T _ {n} $. If

$$ b ( \theta ) = \ {\mathsf E} _ \theta \{ d ( X) \} = \ {\mathsf E} _ \theta \{ T _ {n} \} - g ( \theta ) $$

is the zero vector for all $ \theta \in \Theta $, then one says that $ T _ {n} $ is an unbiased estimator of $ g ( \theta ) $ or that $ T _ {n} $ is free of systematic errors; otherwise, $ T _ {n} $ is said to be biased, and the vector $ b ( \theta ) $ is called the bias or systematic error of the point estimator. The quality of a point estimator can be defined by means of the risk function (cf. Risk of a statistical procedure).

References

[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)

Comments

References

[a1] E.L. Lehmann, "Theory of point estimation" , Wiley (1983)
How to Cite This Entry:
Point estimator. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Point_estimator&oldid=48211
This article was adapted from an original article by M.S. Nikulin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article