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Unformatted text preview: COMPOSITE TESTS AND THE LIKELIHOOD RATIO TEST Lecture XXIV Simple Tests Against a Composite Mathematically, we now can express the tests as testing between H : = against H 1 : 1 , where 1 is a subset of the parameter space. 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. 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 ( 29 ( 29 R X P Q = Definition 9.4.2. Let Q 1 ( ) and Q 2 ( ) 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 1 : 1 if Q 1 ( )= Q 2 ( ) and ( 29 ( 29 ( 29 ( 29 1 2 1 1 2 1 one least at for and all for Q Q Q Q Definition 9.4.3. A test R is the uniformly most...
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 Fall '09
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