factorization criterion
Let $\bm{X}=({X}_{1},\mathrm{\dots},{X}_{n})$ be a random vector whose coordinates^{} are observations, and whose probability (density^{}) function is, $f(\bm{x}\mid \theta )$ where $\theta $ is an unknown parameter. Then a statistic^{} $T(\bm{X})$ for $\theta $ is a sufficient statistic iff $f$ can be expressed as a product of (or factored into) two functions $g,h$, $f=gh$ where $g$ is a function of $T(\bm{X})$ and $\theta $, and $h$ is a function of $\bm{x}$. In symbol, we have
$$f(\bm{x}\mid \theta )=g(T(\bm{X}),\theta )h(\bm{x}).$$ 
Applications.

1.
In view of the above statement, let’s show that the sample mean $\overline{X}$ of $n$ independent^{} observations from a normal distribution^{} $N(\mu ,{\sigma}^{2})$ is a sufficient statistic for the unknown mean $\mu $. Since the ${X}_{i}$’s are independent random variables^{}, then the probability density function^{} $f(\bm{x}\mid \mu )$, being the joint probability density function of each of the ${X}_{i}$, is the product of the individual density functions $f(x\mid \mu )$:
$f(\bm{x}\mid \mu )$ $=$ $\prod _{i=1}^{n}}f(x\mid \mu )={\displaystyle \prod _{i=1}^{n}}{\displaystyle \frac{1}{\sqrt{2\pi {\sigma}^{2}}}}\mathrm{exp}[{\displaystyle \frac{{({x}_{i}\mu )}^{2}}{2{\sigma}^{2}}}]$ (1) $=$ $\frac{1}{\sqrt{{(2\pi )}^{n}{\sigma}^{2n}}}}\mathrm{exp}\left[{\displaystyle \sum _{i=1}^{n}}{\displaystyle \frac{{({x}_{i}\mu )}^{2}}{2{\sigma}^{2}}}\right]$ (2) $=$ $\frac{1}{\sqrt{{(2\pi )}^{n}{\sigma}^{2n}}}}\mathrm{exp}\left[{\displaystyle \frac{1}{2{\sigma}^{2}}}{\displaystyle \sum _{i=1}^{n}}{x}_{i}^{2}\right]\mathrm{exp}\left[{\displaystyle \frac{\mu}{{\sigma}^{2}}}{\displaystyle \sum _{i=1}^{n}}{x}_{i}{\displaystyle \frac{n{\mu}^{2}}{2{\sigma}^{2}}}\right]$ (3) $=$ $h(\bm{x})\mathrm{exp}\left[{\displaystyle \frac{n\mu}{{\sigma}^{2}}}T(\bm{x}){\displaystyle \frac{n{\mu}^{2}}{2{\sigma}^{2}}}\right]$ (4) $=$ $h(\bm{x})g(T(\bm{x}),\mu )$ (5) where $g$ is the last exponential expression and $h$ is the rest of the expression in $(3)$. By the factorization criterion, $T(\bm{X})=\overline{X}$ is a sufficient statistic.

2.
Similarly, the above shows that the sample variance ${s}^{2}$ is not a sufficient statistic for ${\sigma}^{2}$ if $\mu $ is unknown.

3.
But, if $\mu $ is a known constant, then the statistic
$$T({X}_{1},\mathrm{\dots},{X}_{n})=\frac{1}{n1}\sum _{i=1}^{n}{({X}_{i}\mu )}^{2}$$ is sufficient for ${\sigma}^{2}$ by observing in $(2)$ above, and letting $h(\bm{x})=1$ and $g(T,{\sigma}^{2})$ be all of expression $(2)$.
Title  factorization criterion 

Canonical name  FactorizationCriterion 
Date of creation  20130322 15:02:48 
Last modified on  20130322 15:02:48 
Owner  CWoo (3771) 
Last modified by  CWoo (3771) 
Numerical id  4 
Author  CWoo (3771) 
Entry type  Theorem 
Classification  msc 62B05 
Synonym  factorization theorem 
Synonym  FisherNeyman factorization theorem 