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by: Shane Marks


Shane Marks

GPA 3.93


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Class Notes
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This 4 page Class Notes was uploaded by Shane Marks on Monday October 26, 2015. The Class Notes belongs to STAT 703 at University of South Carolina - Columbia taught by Staff in Fall. Since its upload, it has received 7 views. For similar materials see /class/229670/stat-703-university-of-south-carolina-columbia in Statistics at University of South Carolina - Columbia.




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Date Created: 10/26/15
STAT 703J703 February 13th 2007 Lecz ure 7 7 lnstruotor Brian Habing Department of Statistics LeConte 203 Telephone 8037773578 Email habingstatsoedu STAT 702702 BHabing Univ ofSC Today 39 Basic Hypothesis Testing Examples continued 39 NeymanPearson Lemma STAT 702702 BHabing Univ of SC 4 Example 1 Consider a sample of size 1 from a normal distribution with variance 1 Test H01u0 VS HAiu1 at OL005 STAT 702702 BHabing Univ of SC 4 Example 2 Consider a sample of size 1 from a normal distribution with variance 1 Test H0usO vs HApgtO at oc005 ltltltlt M STAT 702702 BHabing Univ ofSC n E For a composite test the significance level or is the maximum supremum of the probabilities of a Type I error over all the possible alternatives V PM K39 null lm ltlt STAT 702702 BHabing Univ of SC E The NeymanPearson Lemma Typically there are several possible tests of HO vs HA for a given level of significance oc How do we select the best in what sense to use Best test A test which has the correct significance level or and is as or more powerful 13 is greater than other test with the same significachVe level oc STAT 702702 BHabing Univ ofSC null lm The NeymanPearson theory shows that a best test exists for simple HO vs simple HA and is based on the ratio of the likelikhood functions and on the two hypotheses ie WK IMHO mix IikltHAgt where ikH is the likelihood function when H is true STAT 702702 BHabing Univ of SC 4 A fo The likelihood ratio ME gives the relative plausibilities of HO and HA Reject HO if the likelihood ratio 7 is small 7 s c where c is chosen to give significance level or STAT 702702 BHabing Univ of SC 4 NeymanPearson Lemma If the likelihood ratio test that rejects H0 in favor of HA when f0x lt has significance level or C Jill 9 then any other test having significance level at most 0i has power less than or egual to the power of the likelihood ratio test le the LRT has highest power among tests with significance level or STAT 702702 BHabing Univ of SC 4 Example Consider a sample of size n from a normal distribution with variance 1 Test H01u0 VS HAiu1 at OL005 ltltltlt M STAT 702702 BHabing Univ ofSC r n 10 a E In some cases we can also show that the test is uniformly most powerful for a composite alternate hypotheses This happens if we can show it is most powerful for everysimple alternate in H A V M STAT 702702 BHabing Univ of SC n 11 ltlt E Consider testing Test H0u0 vs HAugtO and Test H0u0 vs HAu O STAT 702702 BHabing Univ of SC 12


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