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

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 ′...
View Full Document

This note was uploaded on 11/29/2011 for the course MATH 4056 taught by Professor Staff during the Fall '08 term at LSU.

Page1 / 3

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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online