notes-11-03

notes-11-03 - M4056 Hypothesis Testing IV November 3,5 2010...

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Unformatted text preview: M4056 Hypothesis Testing IV. November 3,5, 2010 A. Review I’ll summarize the testing situation in a nutshell. Let θ parametrize a a family of random variables X θ . Let vector X θ be a sample consisting of n random variables that are i.i.d. X θ . 1 Assume that a value of θ is fixed, and sample data vectorx is produced. The statistician makes judgments about θ based upon vectorx . In hypothesis testing, • we assume that the set of all possible values of θ is the disjoint union of two subsets Θ and Θ 1 ; we take H be the “hypothesis” that θ belongs to Θ rather than Θ 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 β ( θ ) := P θ ( vector X ∈ R ). This is the probability of rejection, as a function of the parameter. The size of a test is sup { β ( θ ) | θ ∈ Θ } . 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 θ , 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 β and β ′ , we say T is uniformly more powerful than T ′ if β ≥ β ′ on the set Θ 1 . In other words, for all θ , if H ( θ ) is false, then P θ ( vector X ∈ R ) ≥ P θ ( vector X ∈ R ′ ), i.e., T has greater probability of rejecting H than T ′...
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This note was uploaded on 11/29/2011 for the course MATH 4056 taught by Professor Staff during the Fall '08 term at LSU.

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notes-11-03 - M4056 Hypothesis Testing IV November 3,5 2010...

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