notes-11-03

notes-11-03 - M4056 Hypothesis Testing IV. November 3,5,...

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Unformatted text preview: M4056 Hypothesis Testing IV. November 3,5, 2010 A. Review Ill summarize the testing situation in a nutshell. Let parametrize a a family of random variables X . Let vector X be a sample consisting of n random variables that are i.i.d. X . 1 Assume that a value of is fixed, and sample data vectorx is produced. The statistician makes judgments about based upon vectorx . In hypothesis testing, we assume that the set of all possible values of is the disjoint union of two subsets and 1 ; we take H be the hypothesis that belongs to rather than 1 ; we divide the possible values of vector X into two disjoint sets: A , the acceptance region, and R , the rejection region, thus creating a test of H . The power function of a test is ( ) := P ( vector X R ). This is the probability of rejection, as a function of the parameter. The size of a test is sup { ( ) | } . This is the maximum probability of rejection if H is true. If the size of a test is less than a given number, we say the level of significance is (better than) that number. In practice, a test constructed to meet two criteria. First, a level of significance is given. Second, among tests attaining the desired level of significance, the more powerful tests are sought. Of course, the power varies with , so in general comparing power means comparing two functions. If T and T are two tests determined by rejection regions R and R and having power functions and , we say T is uniformly more powerful than T if on the set 1 . In other words, for all , if H ( ) is false, then P ( vector X R ) P ( vector X R ), i.e., T has greater probability of rejecting H than T...
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notes-11-03 - M4056 Hypothesis Testing IV. November 3,5,...

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