# 409Exam1Aans - STAT 409 Fall 2011 Version A Name ANSWERS ....

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1. Let X 1 , X 2 , … , X n be a random sample from the distribution with probability density function ( ) ( ) ( ) α 1 1 α α ; X x x f - + = , 0 < x < 1, α > – 1. a) (4) Find a sufficient statistic Y = u ( X 1 , X 2 , … , X n ) for α . θ = α ( ) = n i i f 1 X θ X ; = ( ) ( ) ( ) ( ) θ θ 1 1 X 1 1 X 1 1 θ θ - + - + = = = n i i n n i i . By Factorization Theorem, Y 1 = ( ) = - n i i 1 X 1 is a sufficient statistic for θ . Y 2 = ln Y 1 = ( ) = - n i i 1 X 1 ln is also a sufficient statistic for θ . OR f ( x ; θ ) = ( ) ( ) { } 1 1 θ θ ln ln exp + + - x . K ( x ) = ( ) 1 ln x - . Y 2 = ( ) = n i i 1 X K = ( ) = - n i i 1 X 1 ln is a sufficient statistic for θ . Y
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## This note was uploaded on 10/13/2011 for the course STAT 409 taught by Professor Stephanov during the Fall '11 term at University of Illinois at Urbana–Champaign.

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409Exam1Aans - STAT 409 Fall 2011 Version A Name ANSWERS ....

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