10_21_11 - STAT 409 Examples for 10/21/2011 Fall 2011 H : =...

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Unformatted text preview: STAT 409 Examples for 10/21/2011 Fall 2011 H : = vs. H 1 : = 1 . Likelihood Ratio: ( ) ( ) ( ) ,..., , ; ,..., , ; ,..., , 2 1 1 2 1 2 1 L L n n n x x x x x x x x x = . Neyman-Pearson Theorem : C = { ( x 1 , x 2 , , x n ) : ( ) k x x x n ,..., , 2 1 }. ( Reject H if ( ) k x x x n ,..., , 2 1 ) is the best (most powerful) rejection region. 1. Let X 1 , X 2 , , X n be a random sample of size n from a N ( , 2 ) distribution ( 2 is known ). Use the likelihood ratio to find the best rejection region for the test H : = vs. H 1 : = 1 . ( ) ( ) ( ) ( ) ( ) = = -- -- = = n i i n i i n n n x x x x x x x x x x x 1 2 1 2 1 2 2 2 1 1 2 1 2 1 L L 2 1 exp 2 1 2 1 exp 2 1 ,..., , ; ,..., , ; ,..., , = ( ) ( ) [ ] --- = 2 1 exp 1 2 2 1 2 n i i i x x = ( ) ( ) - +- 2 exp 2 1 2 2 2 1 x n n ( ) k x x x n ,..., , 2 1 ( ) x - 1 k 1 < > 1 1 if if c x c x 2. Let X 1 , X 2 , , X n be a random sample of size n from a Poisson distribution with mean . That is, P ( X 1 = k ) = ! k e k- , k = 0, 1, 2, 3, . Consider the test H : = vs. H 1 : = 1 . Show that the best rejection region is given by { ( x 1 , x 2 , , x n ) : = n i i x 1 c } if 1 > , and by { ( x 1 , x 2 , , x n ) : = n i i x 1 c } if 1 < ....
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10_21_11 - STAT 409 Examples for 10/21/2011 Fall 2011 H : =...

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