10_26 - STAT 410 Examples for 10/26/2011 Fall 2011 Def Let...

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STAT 410 Examples for 10/26/2011 Fall 2011 Def Let X 1 , X 2 , … , X n denote random variables with joint p.d.f. or p.m.f. f ( x 1 , x 2 , , x n ; θ ) , which depends on the parameter θ . The statistic Y = u ( X 1 , X 2 , , X n ) is said to be sufficient for θ if the conditional distribution of X 1 , X 2 , … , X n given Y = y is independent of θ for all y . Theorem 1 ( Factorization Theorem ) : Let X 1 , X 2 , … , X n denote random variables with joint p.d.f. or p.m.f. f ( x 1 , x 2 , , x n ; θ ) , which depends on the parameter θ . The statistic Y = u ( X 1 , X 2 , , X n ) is sufficient for θ if and only if f ( x 1 , x 2 , , x n ; θ ) = φ [ u ( x 1 , x 2 , , x n ) ; θ ] h ( x 1 , x 2 , , x n ) , where depends on x 1 , x 2 , , x n only through u ( x 1 , x 2 , , x n ) and h ( x 1 , x 2 , , x n ) does not depend on θ . 1. Let X 1 , X 2 , … , X n be a random sample of size n from a Poisson distribution with mean λ . That is,
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10_26 - STAT 410 Examples for 10/26/2011 Fall 2011 Def Let...

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