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
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.
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 .
|Date of creation||2013-03-22 15:02:48|
|Last modified on||2013-03-22 15:02:48|
|Last modified by||CWoo (3771)|
|Synonym||Fisher-Neyman factorization theorem|