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AMDA-2008-Session 8

# AMDA-2008-Session 8 - Hypothesis Testing and P-values II...

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Hypothesis Testing and P-values – II

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Likelihood Ratio Test 0 0 0 0 1 0 0 to restricted is when mle the is ˆ and mle the is ˆ where ) ˆ ( ) ˆ ( ln 2 ) ( sup ) ( sup ln 2 is statistic ratio likelihood The : H versus : H esting Consider t 0 Θ = = Λ Θ Θ Θ Θ θ θ θ θ θ θ θ θ θ θ θ L L L L
Likelihood Ratio Test . of value observed the is where ) P( is test for the value - p The . of dimension the minus of dimension the is q - r where ~ ) ,..., ( , : H Under statistic. ratio likelihood the be Let )}. ,..., ( ) ,..., ( : { Let ). ,..., , ,..., ( that Suppose 2 q - r 0 2 1 0 0 , 0 1 , 0 1 0 1 1 Λ Θ Θ Λ Θ Λ = = Θ = - + + + λ λ χ χ θ θ θ θ θ θ θ θ θ θ θ q r a n r q r q r q q X X

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Let X 1 ,…,X n be iid N(μ,1). Find the LRT for H 0 : μ=0 versus H 1 :μ ≠ 0 Let X 1 ,…,X n be iid N(μ,σ). Find the LRT for H 0 : (μ,σ)=(0,1) versus H 1 :(μ,σ) ≠ (0,1) Let X 1 ,…,X n be iid Ber(p). Find the LRT for H 0 : p=0.5 versus H 1 :p ≠ 0.5 Let X 1 ,…,X n be iid Poisson(μ). Find the LRT for H 0 : μ ≤ 1 versus H 1 :μ>1 Likelihood Ratio Test
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AMDA-2008-Session 8 - Hypothesis Testing and P-values II...

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