factorization criterion
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.
-
1.
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 :
(1) (2) (3) (4) (5) where is the last exponential expression and is the rest of the expression in . By the factorization criterion, is a sufficient statistic.
-
2.
Similarly, the above shows that the sample variance is not a sufficient statistic for if is unknown.
-
3.
But, if is a known constant, then the statistic
is sufficient for by observing in above, and letting and be all of expression .
Title | factorization criterion |
---|---|
Canonical name | FactorizationCriterion |
Date of creation | 2013-03-22 15:02:48 |
Last modified on | 2013-03-22 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 | Fisher-Neyman factorization theorem |