409Hw09ans

# 409Hw09ans - STAT 409 Fall 2011 Homework #9 (due Friday,...

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STAT 409 Homework #9 Fall 2011 (due Friday, November 11, by 4:00 p.m.) 1. Let λ > 0 and let X 1 , X 2 , … , X n be independent random variables, each with the probability density function f ( x ) = < + 1 0 1 1 λ λ x x x . We wish to test H 0 : λ = 1 vs. H 1 : λ > 1. a) Find a sufficient statistic for λ . f ( x 1 ; λ ) f ( x 2 ; λ ) f ( x n ; λ ) = 1 1 λ λ + = n i i n x . = n i i 1 X is sufficient for λ . OR f ( x ; λ ) = exp { λ ln x + ln λ ln x }. K ( x ) = ln x . = n i i 1 X ln is a sufficient statistic for λ . b) Find a uniformly most powerful rejection region. That is, find a rejection region that is most powerful for testing H 0 : λ = 1 vs. H 1 : λ = λ 1 for all λ 1 > 1. Hint: It should look like “Reject H 0 if Y c or “Reject H 0 if Y c ”, where Y = u ( X 1 , X 2 , … , X n ) is a sufficient statistic for λ . ( ) , ... , , 2 1 λ n x x x = ( ) ( ) , ... , , ; , ... , , ; 1 2 1 2 1 λ L L n n x x x x x x = 1 1 2 1 1 2 2 2 2 1 λ λ λ ... ... λ - - - - - - - - - n n n x x x x x x = n n x x x λ 1 1 2 1 1 λ λ λ ... - - - = 1 1 λ λ 1 - = n i i n x .

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Since λ > 1, ( ) , ... , , 2 1 λ n x x x k = n i i x 1 c . Uniformly most powerful rejection region is given by
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## 409Hw09ans - STAT 409 Fall 2011 Homework #9 (due Friday,...

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