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Unformatted text preview: Minimal Sufficient Statistics Minimal Sufficiency A SS Y 1 = u ( X 1 , , X n ) is called a MSS for if, for any other SS W , Y 1 is a function of W . ? Any invertible function of a MSS is a MSS. ? If MLE is a SS itself, then is a MSS. Theorem (Lehmann-Scheff e) X 1 , , X n has joint pdf f ( x 1 , , x n ; ). A statistic Y 1 = u ( X 1 , , X n ) is a minimal suffi- cient statistic for if the following is true: For any ( x 1 , , x n ) and ( x 1 , , x n ), f ( x 1 , ,x n ; ) f ( x 1 , ,x n ; ) is constant as a function of iff u ( x 1 , , x n ) = u ( x 1 , , x n ). 1 Ancillary Statistics X 1 , , X n f ( x 1 , , x n ; ), a statistic S ( X 1 , , X n ) is called an ancillary statistic for if its distribution does not depend on the parameter . X N ( , 1), S 2 or max X i- min X i is ancillary for ....
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- Fall '07