# Factorization theorem

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