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 00 : H  against 11 : 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 : H against : H if     1 0 2 0 QQ and         1 2 1 1 2 1 for all and for at least one      C. Definition 9.4.3. A test R is the uniformly most power (UMP) test of size (level) for testing : H against : H if     0 PR   and any other test 1 R such that     10 , we have     1 P R P R for any 1 .
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This note was uploaded on 07/18/2011 for the course AEB 6933 taught by Professor Carriker during the Fall '09 term at University of Florida.

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

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