Difference between revisions of "Factorization theorem"
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====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> |
− | + | <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> | |
+ | <TR><TD valign="top">[a1]</TD> <TD valign="top"> D.R. Cox, D.V. Hinkley, "Theoretical statistics" , Chapman & Hall (1974) pp. 21</TD></TR> | ||
+ | </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 |
Factorization theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Factorization_theorem&oldid=55788