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Lecture 24-2007

Lecture 24-2007 - Composite Tests and The Likelihood Ratio...

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1 Composite Tests and The Likelihood Ratio Test Lecture XXIV I. Simple Tests Against a Composite A. Mathematically, we now can express the tests as testing between 0 0 : H against 1 1 : H  , where 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 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 depends on a vector of parameters , we define the power function of the test based on the critical region R by   Q P X R   . 2. Definition 9.4.2. Let 1 Q and 2 Q 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 0 0 : H against 1 1 : H  if 1 0 2 0 Q Q and 1 2 1 1 2 1 for all and for at least one Q Q Q Q       C. Definition 9.4.3. A test R is the uniformly most power (UMP) test of size (level) for testing 0 0 : H against 1 1 : H  if 0 P R   and any other test 1 R such that 1 0 P R   ,

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• Fall '09
• CARRIKER
• Null hypothesis, Hypothesis testing, Statistical hypothesis testing, Likelihood function, Statistical power, likelihood ratio test

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Lecture 24-2007 - Composite Tests and The Likelihood Ratio...

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