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''factorization criterion''
 
''factorization criterion''
  
A theorem in the theory of statistical estimation giving a necessary and sufficient condition for a statistic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038120/f0381201.png" /> to be sufficient for a family of probability distributions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038120/f0381202.png" /> (cf. [[Sufficient statistic|Sufficient statistic]]).
+
A theorem in the theory of statistical estimation giving a necessary and sufficient condition for a statistic $  T $
 +
to be sufficient for a family of probability distributions $  \{ P _  \theta  \} $(
 +
cf. [[Sufficient statistic|Sufficient statistic]]).
 +
 
 +
Let  $  X $
 +
be a random vector taking values in a sample space  $  ( \mathfrak X , {\mathcal B} , P _  \theta  ) $,
 +
$  \theta \in \Theta $,
 +
where the family of probability distributions  $  \{ P _  \theta  \} $
 +
is dominated by some measure  $  \mu $,
 +
and let
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038120/f0381203.png" /> be a random vector taking values in a sample space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038120/f0381204.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038120/f0381205.png" />, where the family of probability distributions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038120/f0381206.png" /> is dominated by some measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038120/f0381207.png" />, and let
+
$$
 +
p ( x;  \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/f/f038/f038120/f0381208.png" /></td> </tr></table>
+
\frac{dP _  \theta  ( x) }{d \mu }
 +
,\ \
 +
\theta \in \Theta .
 +
$$
  
Further, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038120/f0381209.png" /> be a statistic constructed from the observation vector of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038120/f03812010.png" /> and mapping the measurable space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038120/f03812011.png" /> into the measurable space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038120/f03812012.png" />. Under these conditions the following question arises: When is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038120/f03812013.png" /> sufficient for the family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038120/f03812014.png" />? As an answer to this question, the factorization theorem asserts: For a statistic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038120/f03812015.png" /> to be sufficient for a family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038120/f03812016.png" /> that admits sufficient statistics, it is necessary and sufficient that for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038120/f03812017.png" /> the probability density <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038120/f03812018.png" /> can be factorized in the following way:
+
Further, let $  T = T ( X) $
 +
be a statistic constructed from the observation vector of $  X $
 +
and mapping the measurable space $  ( \mathfrak X , {\mathcal B} ) $
 +
into the measurable space $  ( \mathfrak Y , {\mathcal A} ) $.  
 +
Under these conditions the following question arises: When is $  T $
 +
sufficient for the family $  \{ P _  \theta  \} $?  
 +
As an answer to this question, the factorization theorem asserts: For a statistic $  T $
 +
to be sufficient for a family $  \{ P _  \theta  \} $
 +
that admits sufficient statistics, it is necessary and sufficient that for every $  \theta \in \Theta $
 +
the probability density $  p ( x;  \theta ) $
 +
can be factorized in the following way:
  
<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/f/f038/f038120/f03812019.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
+
$$ \tag{* }
 +
p ( x; \theta )  = \
 +
g ( x) h ( T ( x); \theta ),
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038120/f03812020.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038120/f03812021.png" />-measurable function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038120/f03812022.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038120/f03812023.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038120/f03812024.png" />-measurable function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038120/f03812025.png" />. The factorization theorem, beyond giving a criterion for sufficiency, in many cases enables one to determine the concrete form of the sufficient statistic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038120/f03812026.png" /> for which the density <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038120/f03812027.png" /> must factorize by the formula (*). In practice it is usually preferable to deal with the likelihood function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038120/f03812028.png" /> rather than with the density <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038120/f03812029.png" />. In terms of the likelihood function the condition (*) has the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038120/f03812030.png" />, explicitly containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038120/f03812031.png" />.
+
where $  g ( \cdot ) $
 +
is a $  {\mathcal B} $-
 +
measurable function on $  ( \mathfrak X , {\mathcal B} ) $,  
 +
and $  h ( \cdot , \theta ) $
 +
is an $  {\mathcal A} $-
 +
measurable function on $  ( \mathfrak Y , {\mathcal A} ) $.  
 +
The factorization theorem, beyond giving a criterion for sufficiency, in many cases enables one to determine the concrete form of the sufficient statistic $  T $
 +
for which the density $  p ( x;  \theta ) $
 +
must factorize by the formula (*). In practice it is usually preferable to deal with the likelihood function $  L ( \theta ) = p ( X;  \theta ) $
 +
rather than with the density $  p ( x;  \theta ) $.  
 +
In terms of the likelihood function the condition (*) has the form $  L ( \theta ) = g ( X) h ( T;  \theta ) $,  
 +
explicitly containing $  T $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  R.A. Fischer,  "On the mathematical foundations of theoretical statistics"  ''Philos. Trans. Roy. Soc. London Ser. A'' , '''222'''  (1922)  pp. 309–368</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  J. Neyman,  "Su un teorema concernente le cosiddette statistiche sufficienti"  ''Giorn. Istit. Ital. Att.'' , '''6'''  (1935)  pp. 320–334</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  E.L. Lehmann,  "Testing statistical hypotheses" , Wiley  (1959)</TD></TR><TR><TD valign="top">[4]</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">[5]</TD> <TD valign="top">  P.R. Halmos,  L.J. Savage,  "Application of the Radon–Nikodym theorem to the theory of sufficient statistics"  ''Ann. of Math. Statist.'' , '''20'''  (1949)  pp. 225–241</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  R.A. Fischer,  "On the mathematical foundations of theoretical statistics"  ''Philos. Trans. Roy. Soc. London Ser. A'' , '''222'''  (1922)  pp. 309–368</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  J. Neyman,  "Su un teorema concernente le cosiddette statistiche sufficienti"  ''Giorn. Istit. Ital. Att.'' , '''6'''  (1935)  pp. 320–334</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  E.L. Lehmann,  "Testing statistical hypotheses" , Wiley  (1959)</TD></TR><TR><TD valign="top">[4]</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">[5]</TD> <TD valign="top">  P.R. Halmos,  L.J. Savage,  "Application of the Radon–Nikodym theorem to the theory of sufficient statistics"  ''Ann. of Math. Statist.'' , '''20'''  (1949)  pp. 225–241</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  D.R. Cox,  D.V. Hinkley,  "Theoretical statistics" , Chapman &amp; Hall  (1974)  pp. 21</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  D.R. Cox,  D.V. Hinkley,  "Theoretical statistics" , Chapman &amp; Hall  (1974)  pp. 21</TD></TR></table>

Revision as of 19:38, 5 June 2020


factorization criterion

A theorem in the theory of statistical estimation giving a necessary and sufficient condition for a statistic $ T $ to be sufficient for a family of probability distributions $ \{ P _ \theta \} $( cf. Sufficient statistic).

Let $ X $ be a random vector taking values in a sample space $ ( \mathfrak X , {\mathcal B} , P _ \theta ) $, $ \theta \in \Theta $, where the family of probability distributions $ \{ P _ \theta \} $ is dominated by some measure $ \mu $, and let

$$ p ( x; \theta ) = \ \frac{dP _ \theta ( x) }{d \mu } ,\ \ \theta \in \Theta . $$

Further, let $ T = T ( X) $ be a statistic constructed from the observation vector of $ X $ and mapping the measurable space $ ( \mathfrak X , {\mathcal B} ) $ into the measurable space $ ( \mathfrak Y , {\mathcal A} ) $. Under these conditions the following question arises: When is $ T $ sufficient for the family $ \{ P _ \theta \} $? As an answer to this question, the factorization theorem asserts: For a statistic $ T $ to be sufficient for a family $ \{ P _ \theta \} $ that admits sufficient statistics, it is necessary and sufficient that for every $ \theta \in \Theta $ the probability density $ p ( x; \theta ) $ can be factorized in the following way:

$$ \tag{* } p ( x; \theta ) = \ g ( x) h ( T ( x); \theta ), $$

where $ g ( \cdot ) $ is a $ {\mathcal B} $- measurable function on $ ( \mathfrak X , {\mathcal B} ) $, and $ h ( \cdot , \theta ) $ is an $ {\mathcal A} $- measurable function on $ ( \mathfrak Y , {\mathcal A} ) $. The factorization theorem, beyond giving a criterion for sufficiency, in many cases enables one to determine the concrete form of the sufficient statistic $ T $ for which the density $ p ( x; \theta ) $ must factorize by the formula (*). In practice it is usually preferable to deal with the likelihood function $ L ( \theta ) = p ( X; \theta ) $ rather than with the density $ p ( x; \theta ) $. In terms of the likelihood function the condition (*) has the form $ L ( \theta ) = g ( X) h ( T; \theta ) $, explicitly containing $ T $.

References

[1] R.A. Fischer, "On the mathematical foundations of theoretical statistics" Philos. Trans. Roy. Soc. London Ser. A , 222 (1922) pp. 309–368
[2] J. Neyman, "Su un teorema concernente le cosiddette statistiche sufficienti" Giorn. Istit. Ital. Att. , 6 (1935) pp. 320–334
[3] E.L. Lehmann, "Testing statistical hypotheses" , Wiley (1959)
[4] I.A. Ibragimov, R.Z. [R.Z. Khas'minskii] Has'minskii, "Statistical estimation: asymptotic theory" , Springer (1981) (Translated from Russian)
[5] P.R. Halmos, L.J. Savage, "Application of the Radon–Nikodym theorem to the theory of sufficient statistics" Ann. of Math. Statist. , 20 (1949) pp. 225–241

Comments

References

[a1] D.R. Cox, D.V. Hinkley, "Theoretical statistics" , Chapman & Hall (1974) pp. 21
How to Cite This Entry:
Factorization theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Factorization_theorem&oldid=46900
This article was adapted from an original article by M.S. Nikulin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article