Lecture13

# Lecture13 - April 9th Likelihood Ratio Test(Another method...

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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 μ σ where 2 σ is known. We wish to test : H μ μ = versus : a H μ μ ≠ at the significance level α . Please derive the test using the likelihood ratio method. Solution. 1. Write down your parameter space under H { } : ϖ μ μ μ = = 2. Write down the unrestricted/original parameter space. { } : R μ μ Ω = ∈ 3. Write down the likelihood (of the data) 1 2 ( , , , ; ) n L f x x x μ = L 2 2 ( ) 2 1 1 1 ( ; ) 2 i x n n i i i f x e μ σ μ πσ-- = = = = ∏ ∏ 2 1 2 ( ) 2 2 2 (2 ) n i i x n e μ σ πσ =--- ∑ = ⋅ 4. Write down your log-likelihood. 2 2 1 2 ( ) ln ln(2 ) 2 2 n i i x n l L μ πσ σ =- = = -- ∑ 5. Find your MLE under ϖ and plug it in to your L to obtain max L ϖ 1 2 1 2 ( ) 2 2 2 1 2 max ( , , , ; ) (2 ) n i i x n n L L x x x e μ σ ϖ μ πσ =--- ∑ = = ⋅ L 6. Find the MLE(s) under Ω and plug in to your L to obtain max L Ω 1 2 2 ( ) ln ˆ 2 n i i x d L X d μ μ μ σ =- = ⇒ = ⇒ = ∑ 2 1 2 ( ) 2 2 2 1 2 ˆ max ( , , , ; ) (2 ) n i i x x n n L L x x x e σ μ πσ =--- Ω ∑ = = ⋅ L 7. Get the likelihood ratio 2 2 2 1 2 1 1 2 2 1 2 ( ) (...
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## This note was uploaded on 11/14/2010 for the course AMS 312 taught by Professor Zhu,w during the Spring '08 term at SUNY Stony Brook.

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Lecture13 - April 9th Likelihood Ratio Test(Another method...

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