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Unformatted text preview: Composite Tests and the Likelihood Ratio Test: Lecture XXIV Charles B. Moss October 28, 2010 Charles B. Moss () Composite and Likelihood Ratio October 28, 2010 1 / 11 1 Simple Tests against a Composite 2 Composite against Composite Charles B. Moss () Composite and Likelihood Ratio October 28, 2010 2 / 11 Simple Tests against a Composite Mathematically, we now can express the tests as testing between H : θ = θ against H 1 : θ ∈ Θ 1 , where Θ 1 is a subset of the parameter space. 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. I Definition 9.4.1. If the distribution of the sample X depends on a vector of parameters θ , we define the power function of the test based on the critical region R by Q ( θ ) = P ( X ∈ R  θ ) ( 1 ) I Definition 9.4.2. Let Q 1 ( θ ) and Q 2 ( θ ) be the power functions of two tests respectively. Then we say that the first test is uniformly better (or uniformly most powerful) than the second in testing H : θ = θ against H 1 : θ ∈ Θ 1 if Q 1 ( θ ) = Q 2 ( θ ) and Q 1 ( θ ) ≥ Q 2 ( θ ) for all θ ∈ Θ 1 and Q 1 ( θ ) > Q 2 ( θ ) for at least one...
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This note was uploaded on 07/15/2011 for the course AEB 6180 taught by Professor Staff during the Spring '10 term at University of Florida.
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