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factorization criterion
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(Theorem)
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Let
 be a random vector whose coordinates are observations, and whose probability (density) function is,
 where  is an unknown parameter. Then a statistic
 for  is a sufficient statistic iff  can be expressed as a product of (or factored into) two functions  ,  where  is a function of
 and  , and  is a function of
 . In symbol, we have
Applications.
- In view of the above statement, let's show that the sample mean
 of  independent observations from a normal distribution
 is a sufficient statistic for the unknown mean  . Since the  's are independent random variables, then the probability density function
 , being the joint probability density function of each of the  , is the product of the individual density functions
 :
where  is the last exponential expression and  is the rest of the expression in  . By the factorization criterion,
 is a sufficient statistic.
- Similarly, the above shows that the sample variance
 is not a sufficient statistic for  if  is unknown.
- But, if
 is a known constant, then the statistic
is sufficient for  by observing in  above, and letting
 and
 be all of expression  .
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"factorization criterion" is owned by CWoo.
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(view preamble)
| Other names: |
factorization theorem, Fisher-Neyman factorization theorem |
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Cross-references: sufficient, sample variance, expression, exponential, probability density function, random variables, mean, normal distribution, independent, sample mean, applications, product, iff, sufficient statistic, statistic, parameter, function, density, observations, coordinates, random vector
There is 1 reference to this entry.
This is version 1 of factorization criterion, born on 2005-02-16.
Object id is 6761, canonical name is FactorizationCriterion.
Accessed 4213 times total.
Classification:
| AMS MSC: | 62B05 (Statistics :: Sufficiency and information :: Sufficient statistics and fields) |
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Pending Errata and Addenda
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![$\displaystyle \prod_{i=1}^n f(x\mid\mu)= \prod_{i=1}^n \frac{1}{\sqrt{2\pi\sigma^2}}\exp\Big[-\frac{(x_i-\mu)^2}{2\sigma^2}\Big]$](http://images.planetmath.org:8080/cache/objects/6761/l2h/img25.png "$\displaystyle \prod_{i=1}^n f(x\mid\mu)= \prod_{i=1}^n \frac{1}{\sqrt{2\pi\sigma^2}}\exp\Big[-\frac{(x_i-\mu)^2}{2\sigma^2}\Big]$")
![$\displaystyle \frac{1}{\sqrt{(2\pi)^n\sigma^{2n}}}\exp\Big [\sum_{i=1}^{n}-\frac{(x_i-\mu)^2}{2\sigma^2}\Big]$](http://images.planetmath.org:8080/cache/objects/6761/l2h/img27.png "$\displaystyle \frac{1}{\sqrt{(2\pi)^n\sigma^{2n}}}\exp\Big [\sum_{i=1}^{n}-\frac{(x_i-\mu)^2}{2\sigma^2}\Big]$")
![$\displaystyle \frac{1}{\sqrt{(2\pi)^n\sigma^{2n}}}\exp\Big [\frac{-1}{2\sigma^2... ...ig] \exp\Big[\frac{\mu}{\sigma^2}\sum_{i=1}^n x_i-\frac{n\mu^2}{2\sigma^2}\Big]$](http://images.planetmath.org:8080/cache/objects/6761/l2h/img29.png "$\displaystyle \frac{1}{\sqrt{(2\pi)^n\sigma^{2n}}}\exp\Big [\frac{-1}{2\sigma^2... ...ig] \exp\Big[\frac{\mu}{\sigma^2}\sum_{i=1}^n x_i-\frac{n\mu^2}{2\sigma^2}\Big]$")
![$\displaystyle h(\boldsymbol{x}) \exp\Big[\frac{n\mu}{\sigma^2}T(\boldsymbol{x})-\frac{n\mu^2}{2\sigma^2}\Big]$](http://images.planetmath.org:8080/cache/objects/6761/l2h/img31.png "$\displaystyle h(\boldsymbol{x}) \exp\Big[\frac{n\mu}{\sigma^2}T(\boldsymbol{x})-\frac{n\mu^2}{2\sigma^2}\Big]$")
