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100BHWFS - STAT 100B Homework VII and VIII Solutions...

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STAT 100B Homework VII and VIII Solutions Solutions to Homework VII Prove the optimality of the likelihood ratio test. A: Suppose we want to test H 0 : X p 0 ( x ) versus H 1 : X p 1 ( x ). Given X = x , the decision rule of the likelihood ratio test is to reject H 0 if p 1 ( x ) /p 0 ( x ) > C where C is a cut-off value. For any decision rule, there are two types of errors: Truth Decision Type I error (false positive) H 0 true reject H 0 Type II error(false negative) H 0 false accept H 0 The likelihood ratio test is the most powerful test in the sense that for fixed type I error, the likelihood ratio test has the smallest type II error. Here is an illustrative proof. In the figure below, the plot on top illustrates acceptance region and rejection region of the likelihood ratio test and the corresponding type I and type II errors. Type I Type II Type II reject H 0 Accept Accept - - P 1 ( X ) CP 0 ( X ) - - reject H 0 Accept Accept - - 1 2 3 4 Type II Type II Type I If we shift the rejection region as in the plot at the bottom. Then new type I error = old type I error - area 1 + area 2 + area 3.
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