10_24_11ans - STAT 409 Examples for Fall 2011 H θ = θ vs...

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Unformatted text preview: STAT 409 Examples for 10/24/2011 Fall 2011 H : θ = θ vs. H 1 : θ = θ 1 . Likelihood Ratio: ( ) ( ) ( ) ,..., , ; ,..., , ; ,..., , 2 1 1 2 1 2 1 L L λ n n n x x x x x x x x x θ θ = . Neyman-Pearson Theorem : C = { ( x 1 , x 2 , … , x n ) : ( ) k x x x n ≤ ,..., , 2 1 λ }. ( “ Reject H if ( ) k x x x n ≤ ,..., , 2 1 λ ” ) is the best (most powerful) rejection region. Let X 1 , X 2 , … , X n be a random sample from the distribution with probability density function ( ) 2 3 2 λ 2 ; λ λ x e f x x- = x > 0 λ > 0. We wish to test H : λ = 5 vs. H 1 : λ = 3. a) Find the form of the most powerful rejection region. Reject H if ( ) ( ) ( ) ∏ ∏ = = -- ⋅ ⋅ = = n i x i n i x i n n n i i e e x x x x x x x x x x x 1 3 3 2 1 5 3 2 2 1 1 2 1 2 1 2 2 3 2 5 2 ,..., , ; ,..., , ; ,..., , H L H L λ ≤ k . Since ( ) - ∑ = ⋅ ⋅ = n i i n n x x x...
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10_24_11ans - STAT 409 Examples for Fall 2011 H θ = θ vs...

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