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Difference between revisions of "Factorization theorem"

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<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>
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<table>
 
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<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>
====Comments====
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<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>
 
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<TR><TD valign="top">[3]</TD> <TD valign="top">  E.L. Lehmann,  "Testing statistical hypotheses" , Wiley  (1959)</TD></TR>
====References====
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<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>
<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>
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<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>
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<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>
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</table>

Latest revision as of 18:39, 23 May 2024


factorization criterion

A theorem in the theory of statistical estimation giving a necessary and sufficient condition for a statistic 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
[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=55788
This article was adapted from an original article by M.S. Nikulin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article