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Unformatted text preview: Lecture 22 22.1 One sided hypotheses continued. It remains to prove the second part of the Theorem from last lecture. Namely, we have to show that for any K ( * , ) ( , ) for > . Let us take > and consider two simple hypotheses h 1 : = and h 2 : = . Let us find the most powerful test with error of type one equal to . We know that if we can find a threshold b such that f ( X | ) f ( X | ) < b = then the following test will be the most powerful test with error of type 1 equal to : = ( h 1 : f ( X | ) f ( X | ) b h 2 : f ( X | ) f ( X | ) < b But the monotone likelihood ratio implies that f ( X | ) f ( X | ) < b f ( X | ) f ( X | ) > 1 b V ( T, , ) > 1 b and, since now > , the function V ( T, , ) is strictly increasing in T. Therefore, we can solve this inequality for T and get that T > c b for some c b ....
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This note was uploaded on 10/11/2009 for the course STATISTICS 18.443 taught by Professor Dmitrypanchenko during the Spring '09 term at MIT.
- Spring '09