Factorization theorem

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 (cf. Sufficient statistic).

Let be a random vector taking values in a sample space , , where the family of probability distributions is dominated by some measure , and let

Further, let be a statistic constructed from the observation vector of and mapping the measurable space into the measurable space . Under these conditions the following question arises: When is sufficient for the family ? As an answer to this question, the factorization theorem asserts: For a statistic to be sufficient for a family that admits sufficient statistics, it is necessary and sufficient that for every the probability density can be factorized in the following way:

(*)

where is a -measurable function on , and is an -measurable function on . The factorization theorem, beyond giving a criterion for sufficiency, in many cases enables one to determine the concrete form of the sufficient statistic for which the density must factorize by the formula (*). In practice it is usually preferable to deal with the likelihood function rather than with the density . In terms of the likelihood function the condition (*) has the form , explicitly containing .

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

References

[a1]	D.R. Cox, D.V. Hinkley, "Theoretical statistics" , Chapman & Hall (1974) pp. 21