Namespaces
Variants
Actions

Difference between revisions of "Minimal sufficient statistic"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (tex encoded by computer)
 
Line 1: Line 1:
A statistic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063910/m0639101.png" /> which is a [[Sufficient statistic|sufficient statistic]] for a family of distributions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063910/m0639102.png" /> and is such that for any other sufficient statistic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063910/m0639103.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063910/m0639104.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063910/m0639105.png" /> is some measurable function. A sufficient statistic is minimal if and only if the sufficient <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063910/m0639106.png" />-algebra it generates is minimal, that is, is contained in any other sufficient <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063910/m0639107.png" />-algebra.
+
<!--
 +
m0639101.png
 +
$#A+1 = 26 n = 0
 +
$#C+1 = 26 : ~/encyclopedia/old_files/data/M063/M.0603910 Minimal sufficient statistic
 +
Automatically converted into TeX, above some diagnostics.
 +
Please remove this comment and the {{TEX|auto}} line below,
 +
if TeX found to be correct.
 +
-->
  
The notion of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063910/m0639109.png" />-minimal sufficient statistic (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063910/m06391011.png" />-algebra) is also used. A sufficient <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063910/m06391012.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063910/m06391013.png" /> (and the corresponding statistic) is called <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063910/m06391014.png" />-minimal if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063910/m06391015.png" /> is contained in the completion <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063910/m06391016.png" />, relative to the family of distributions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063910/m06391017.png" />, of any sufficient <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063910/m06391018.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063910/m06391019.png" />. If the family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063910/m06391020.png" /> is dominated by a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063910/m06391021.png" />-finite measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063910/m06391022.png" />, then the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063910/m06391023.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063910/m06391024.png" /> generated by the family of densities
+
{{TEX|auto}}
 +
{{TEX|done}}
  
<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/m/m063/m063910/m06391025.png" /></td> </tr></table>
+
A statistic  $  X $
 +
which is a [[Sufficient statistic|sufficient statistic]] for a family of distributions  $  {\mathcal P} = \{ { {\mathsf P} _  \theta  } : {\theta \in \Theta } \} $
 +
and is such that for any other sufficient statistic  $  Y $,
 +
$  X = g ( Y ) $,
 +
where  $  g $
 +
is some measurable function. A sufficient statistic is minimal if and only if the sufficient  $  \sigma $-
 +
algebra it generates is minimal, that is, is contained in any other sufficient  $  \sigma $-
 +
algebra.
  
is sufficient and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063910/m06391026.png" />-minimal.
+
The notion of a  $  {\mathcal P} $-
 +
minimal sufficient statistic (or  $  \sigma $-
 +
algebra) is also used. A sufficient $  \sigma $-
 +
algebra  $  {\mathcal B} _ {0} $(
 +
and the corresponding statistic) is called  $  {\mathcal P} $-
 +
minimal if  $  {\mathcal B} _ {0} $
 +
is contained in the completion  $  \overline{ {\mathcal B} }\; $,
 +
relative to the family of distributions  $  {\mathcal P} $,
 +
of any sufficient  $  \sigma $-
 +
algebra  $  {\mathcal B} $.  
 +
If the family  $  {\mathcal P} $
 +
is dominated by a  $  \sigma $-
 +
finite measure  $  \mu $,
 +
then the  $  \sigma $-
 +
algebra  $  {\mathcal B} _ {0} $
 +
generated by the family of densities
  
A general example of a minimal sufficient statistic is given by the canonical statistic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063910/m06391027.png" /> of an exponential family
+
$$
 +
\left \{ {
 +
p _  \theta  ( \omega ) =  
 +
\frac{d p }{d \mu }
  
<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/m/m063/m063910/m06391028.png" /></td> </tr></table>
+
( \omega ) } : {\theta \in \Theta } \right \}
 +
$$
 +
 
 +
is sufficient and  $  {\mathcal P} $-
 +
minimal.
 +
 
 +
A general example of a minimal sufficient statistic is given by the canonical statistic  $  T = ( T _ {1} \dots T _ {n} ) $
 +
of an exponential family
 +
 
 +
$$
 +
p _  \theta  ( \omega )  = \
 +
C ( \theta )  \mathop{\rm exp} \
 +
\sum _ { j } Q _ {j} ( \theta ) T _ {j} ( \omega ).
 +
$$
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J.-R. Barra,  "Mathematical bases of statistics" , Acad. Press  (1981)  (Translated from French)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  L. Schmetterer,  "Introduction to mathematical statistics" , Springer  (1974)  (Translated from German)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J.-R. Barra,  "Mathematical bases of statistics" , Acad. Press  (1981)  (Translated from French)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  L. Schmetterer,  "Introduction to mathematical statistics" , Springer  (1974)  (Translated from German)</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><TR><TD valign="top">[a2]</TD> <TD valign="top">  E.L. Lehmann,  "Testing statistical hypotheses" , Wiley  (1986)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E.L. Lehmann,  "Theory of point estimation" , Wiley  (1983)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  E.L. Lehmann,  "Testing statistical hypotheses" , Wiley  (1986)</TD></TR></table>

Latest revision as of 08:00, 6 June 2020


A statistic $ X $ which is a sufficient statistic for a family of distributions $ {\mathcal P} = \{ { {\mathsf P} _ \theta } : {\theta \in \Theta } \} $ and is such that for any other sufficient statistic $ Y $, $ X = g ( Y ) $, where $ g $ is some measurable function. A sufficient statistic is minimal if and only if the sufficient $ \sigma $- algebra it generates is minimal, that is, is contained in any other sufficient $ \sigma $- algebra.

The notion of a $ {\mathcal P} $- minimal sufficient statistic (or $ \sigma $- algebra) is also used. A sufficient $ \sigma $- algebra $ {\mathcal B} _ {0} $( and the corresponding statistic) is called $ {\mathcal P} $- minimal if $ {\mathcal B} _ {0} $ is contained in the completion $ \overline{ {\mathcal B} }\; $, relative to the family of distributions $ {\mathcal P} $, of any sufficient $ \sigma $- algebra $ {\mathcal B} $. If the family $ {\mathcal P} $ is dominated by a $ \sigma $- finite measure $ \mu $, then the $ \sigma $- algebra $ {\mathcal B} _ {0} $ generated by the family of densities

$$ \left \{ { p _ \theta ( \omega ) = \frac{d p }{d \mu } ( \omega ) } : {\theta \in \Theta } \right \} $$

is sufficient and $ {\mathcal P} $- minimal.

A general example of a minimal sufficient statistic is given by the canonical statistic $ T = ( T _ {1} \dots T _ {n} ) $ of an exponential family

$$ p _ \theta ( \omega ) = \ C ( \theta ) \mathop{\rm exp} \ \sum _ { j } Q _ {j} ( \theta ) T _ {j} ( \omega ). $$

References

[1] J.-R. Barra, "Mathematical bases of statistics" , Acad. Press (1981) (Translated from French)
[2] L. Schmetterer, "Introduction to mathematical statistics" , Springer (1974) (Translated from German)

Comments

References

[a1] E.L. Lehmann, "Theory of point estimation" , Wiley (1983)
[a2] E.L. Lehmann, "Testing statistical hypotheses" , Wiley (1986)
How to Cite This Entry:
Minimal sufficient statistic. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Minimal_sufficient_statistic&oldid=15453
This article was adapted from an original article by A.S. Kholevo (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article