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Unformatted text preview: April 9th Likelihood Ratio Test (Another method to construct a test) Example 1. Let 1 2 , , n X X X L be a random sample from 2 ( , ) N where 2 is known. We wish to test : H = versus : a H at the significance level . Please derive the test using the likelihood ratio method. Solution. 1. Write down your parameter space under H { } : = = 2. Write down the unrestricted/original parameter space. { } : R = 3. Write down the likelihood (of the data) 1 2 ( , , , ; ) n L f x x x = L 2 2 ( ) 2 1 1 1 ( ; ) 2 i x n n i i i f x e  = = = = 2 1 2 ( ) 2 2 2 (2 ) n i i x n e = = 4. Write down your loglikelihood. 2 2 1 2 ( ) ln ln(2 ) 2 2 n i i x n l L = = =  5. Find your MLE under and plug it in to your L to obtain max L 1 2 1 2 ( ) 2 2 2 1 2 max ( , , , ; ) (2 ) n i i x n n L L x x x e = = = L 6. Find the MLE(s) under and plug in to your L to obtain max L 1 2 2 ( ) ln 2 n i i x d L X d = = = = 2 1 2 ( ) 2 2 2 1 2 max ( , , , ; ) (2 ) n i i x x n n L L x x x e = = = L 7. Get the likelihood ratio 2 2 2 1 2 1 1 2 2 1 2 ( ) (...
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 Spring '08
 Zhu,W

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