This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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...
View Full
Document
 Fall '08
 Staff
 Statistics

Click to edit the document details