# post14 - Exponential Family A family of pdf's with...

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Exponential Family A family of pdf’s with parameter θ , { f ( x ; θ ): θ Ω } , is called an exponential family if f can be written as f ( x ; θ )= ( exp( p ( θ ) K ( x )+ S ( x )+ q ( θ )) if a<x<b 0 o.w. ? the support of f ,( a, b ) should not depend on θ . X 1 , ··· ,X n : a random sample with a pdf f ( x ; θ )= ( exp( p ( θ ) K ( x )+ S ( x )+ q ( θ )) if a<x<b 0 o.w. Then Y 1 = n i =1 K ( X i ) is suﬃcient for θ . 1 Completeness and Uniqueness Completeness A random variable Z (continuous or discrete) has a pdf in { h ( z ; θ ); θ Ω } .I f E ( u ( Z )) = 0 , θ Ω requires u ( Z )=0wp1fo reachpd f h ( z ; θ ) Ω, then the family { h ( z ; θ ); θ Ω } is called a complete family of pdfs. ? Completeness is a property of a family of pdfs. ? We simply say that Y 1 is a complete S.S. for θ ,i f Y 1 is a SS for θ and the family of { g 1 ( y 1 ; θ ): θ Ω

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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.

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post14 - Exponential Family A family of pdf's with...

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