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Unformatted text preview: Properties of SS Theorem X 1 , , X n is a random sample from f ( x ; ) , . If a sufficient statistic Y 1 = u 1 ( X 1 , , X n ) for exists and the MLE of exists uniquely, then is a function of Y 1 . Example : X 1 , , X n : a random sample from Bernoulli( ), 0 < < 1 X i is sufficient for . The unique MLE is X . Example : X 1 , , X n : a random sample N ( , 2 ), 2 > 0 known,- < < X is sufficient for , X is the unique MLE of . 1 Example : X 1 , , X n : a random sample from Poisson( ). f ( x 1 , , x n ; ) = n Y i =1 x i e- x i ! = x i e- n n Y i =1 x i ! - 1 By factorization theorem, Y 1 = X i is suffi- cient for . Is X also sufficient for ? 2 SSs are not unique Any invertible function of a sufficient statis- tic is also a sufficient statistic. Why?...
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This note was uploaded on 09/10/2009 for the course STATS 517 taught by Professor Song during the Fall '07 term at Purdue University-West Lafayette.
- Fall '07