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Lecture24-2010

# Lecture24-2010 - Composite Tests and the Likelihood Ratio...

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Composite Tests and the Likelihood Ratio Test Charles B. Moss August 3, 2010 I. Simple Tests against a Composite A. Mathematically, we now can express the tests as testing between H 0 : θ = θ 0 against H 1 : θ Θ 1 , where Θ 1 is a subset of the parameter space. B. Given this specification, we must modify our definition of the power of the test because the β value (the probability of accepting the null hypothesis when it is false) is not unique. In this regard, it is useful to develop the power function. 1. Definition 9.4.1. If the distribution of the sample X de- pends on a vector of parameters θ , we define the power func- tion of the test based on the critical region R by Q ( θ ) = P ( X R | θ ) (1) 2. Definition 9.4.2. Let Q 1 ( θ ) and Q 2 ( θ ) be the power func- tions of two tests respectively. Then we say that the first test is uniformly better (or uniformly most powerful) than the second in testing H 0 : θ = θ 0 against H 1 : θ Θ 1 if Q 1 ( θ 0 ) = Q 2 ( θ 0 ) and Q 1 ( θ ) Q 2 ( θ ) for all θ Θ 1 and Q 1 ( θ ) > Q 2 ( θ ) for at least one θ Θ 1 .

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• Spring '10
• Staff
• Null hypothesis, Hypothesis testing, Statistical hypothesis testing, Likelihood function, Statistical power, likelihood ratio test

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Lecture24-2010 - Composite Tests and the Likelihood Ratio...

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