post16 - Minimal Sufficient Statistics Minimal Sufficiency...

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

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post16 - Minimal Sufficient Statistics Minimal Sufficiency...

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