409Hw11ans - STAT 409 Fall 2009 Homework #11 (due Friday,...

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STAT 409 Homework #11 Fall 2009 (due Friday, November 13, 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 x 1 is sufficient 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 . ( 29 , ... , , 2 1 λ n x x x = ( 29 ( 29 , ... , , ; , ... , , ; 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 . Since λ > 1, ( 29 , ... , , 2 1 λ n x x x k = n i i x 1 c .
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Uniformly most powerful rejection region is given by C = { ( x 1 , x 2 , … , x n ) : = n i i x 1 c }. 2. 1. (continued) Let X 1 , X 2 be a random sample of size n = 2 from a probability distribution with p.d.f. f ( x ). c)
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409Hw11ans - STAT 409 Fall 2009 Homework #11 (due Friday,...

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