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Unformatted text preview: STAT 100B Homework VI Solution Problem 1: Suppose we want to test hypotheses H : X p ( x ) versus H 1 : X p 1 ( x ). Suppose our decision rule is to reject H if p 1 ( X ) /p ( X ) > C . (1) Show that this decision rule is the most powerful in the sense that among all the decision rules with the same type I error, it has the minimum type II error. A: Please see the picture in the lecture note. (2) Suppose the prior probability that H 1 is true is . Suppose that the loss due to type I error is l 1 , and the loss due to type II error is l 2 . Show that the optimal decision rule is to reject H if p 1 ( X ) /p ( X ) > C , where C is determined by , l 1 and l 2 . A: The posterior probability of H 1 is p ( H 1 | X ) = p 1 ( X ) / ( p 1 ( X ) + (1- ) p ( X )) . The posterior probability of H is p ( H | X ) = (1- ) p ( X ) / ( p 1 ( X ) + (1- ) p ( X )) . Given X , if we reject H , the risk or the expected loss is p ( H | X ) l 1 . If we accept H , the risk is p ( H...
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This note was uploaded on 03/30/2011 for the course STAT 100B taught by Professor Wu during the Winter '11 term at UCLA.
- Winter '11