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post7 - Best Critical Region Certain Best Tests Methods of...

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Certain Best Tests Methods of constructing good tests simple H 0 : θ = θ 0 vs. simple H 1 : θ = θ 00 Best tests Best critical regions X 1 , · · · , X n : random sample from f ( x ; θ ) What is the best critical region? 1 Best Critical Region C is a subset of A . Then C is called a best critical region of size α for testing H 0 : θ = θ 0 against H 1 : θ = θ 00 if, for every subset A ∈ A for which Pr [( X 1 , · · · , X n ) A ; H 0 ] = α, (a) Pr [( X 1 , · · · , X n ) C ; H 0 ] = α, (b) Pr [( X 1 , · · · , X n ) C ; H 1 ] Pr [( X 1 , · · · , X n ) A ; H 1 ] 2 Test Statistics: Likelihood Ratios Test Statistic: a function (statistic) of the sample data on which the decision is to be based. Likelihood ratio statistic: L ( θ 0 ) L ( θ 00 ) If this ratio is “small”, then we reject H 0 . How “small” is small? 3 Neyman-Pearson Theorem (N-P Lemma) X 1 , · · · , X n is a random sample from f ( x ; θ ). L ( θ ; x 1 , · · · , x n ) = f ( x 1 ; θ ) f ( x 2 ; θ ) · · · f ( x n ; θ ) Ω = { θ 0 , θ 00 } and k > 0 Let C be a subset of A such that (a) L ( θ 0 ; x 1 , ··· ,x n ) L ( θ 00 ; x 1 , ··· ,x n ) k , ( x 1 ,
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  • Fall '07
  • Song
  • Null hypothesis, Statistical hypothesis testing, Statistical theory, Likelihood principle, Uniformly most powerful test

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