07_22 - STAT 410 Examples for 07/22/2009 Spring 2009 Def...

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Unformatted text preview: STAT 410 Examples for 07/22/2009 Spring 2009 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 ; ) , which depends on the parameter . The statistic Y = u ( X 1 , X 2 , , X n ) is said to be sufficient for if the conditional distribution of X 1 , X 2 , , X n given Y = y is independent of 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 ; ) , which depends on the parameter . The statistic Y = u ( X 1 , X 2 , , X n ) is sufficient for if and only if f ( x 1 , x 2 , , x n ; ) = [ u ( x 1 , x 2 , , x n ) ; ] 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 . 1. Let X 1 , X 2 , , X n be a random sample of size n from a Poisson distribution with mean . That is,...
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07_22 - STAT 410 Examples for 07/22/2009 Spring 2009 Def...

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