post9 - Likelihood Ratio Tests Composite versus Composite...

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Likelihood Ratio Tests Composite versus Composite H 0 : θ Θ 0 vs. H 1 : θ Θ 1 Ω=Θ 0 Θ 1 Extend the idea of N-P Lemma, which is based on the likelihood ratio L ( θ 0 ) L ( θ 00 ) . Likelihood ratio statistic λ ( x 1 , ··· ,x n )= sup θ Θ 0 L ( θ ) sup θ Ω L ( θ ) C = { ( x 1 , ··· ,x n ):0 λ ( x 1 , ··· ,x n ) λ 0 } where λ 0 is chosen so that Pr ( λ ( x 1 , ··· ,x n ) λ 0 ; H 0 )= α. 1 Example : X N ( θ 1 2 ) , Ω= { ( θ 1 2 ): -∞ < θ 1 < , 0 2 < ∞} H 0 : θ 1 =0vs. H 1 : θ 1 6 =0 Θ 0 def = { ( θ 1 2 ): θ 1 =0 , 0 2 < ∞} ⊂ Ω. Then H 0 :( θ 1 2 ) Θ 0 vs. H 1 : not H 0 L 0 )= L (0 2 ; x 1 , ··· ,x n ) L (Ω) = L ( θ 1 2 ; x 1 , ··· ,x n ) L ( ˆ Θ 0 )= L ( ˆ Ω) = 2 λ = L ( ˆ Θ 0 ) L ( ˆ Ω) =( 1 1+ n ¯ x 2 ( x i - ¯ x ) 2 ) n/ 2 λ 0 λ λ 0 n | ¯ x | q ( x i - ¯ x ) 2 / ( n - 1) c C = { n | ¯ x | q ( x i - ¯ x ) 2
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This note was uploaded on 09/10/2009 for the course STATS 517 taught by Professor Song during the Fall '07 term at Purdue.

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post9 - Likelihood Ratio Tests Composite versus Composite...

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