∙ as we will see ump tests generally do not exist

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Unformatted text preview: ∙ As we will see, UMP tests generally do not exist against two-sided alternatives: usually some tests have more power against some alternatives while others have more power against other alternatives. ∙ In such cases, it is helpful to further restrict the class of tests. 45 Unbiased Test ∙ What would be a sensible way to restrict a class of tests in order to find a UMP test within the class? DEFINITION : A test of size is unbiased if P Reject H | ≤ , all ∈ Θ P Reject H | ≥ , all ∈ Θ 1 ∙ The first inequality essentially defines the test to be of size . ∙ The second inequality ensures that the probability of rejecting the null when it is false is at least as large as the probability of rejecting the null when it is true. 46 ∙ If a test is UMP in the class of unbiased tests, the test is said to be uniformly most powerful unbiased ( UMPU ). ∙ For certain problems, UMPU tests are known to exist even if UMP tests do not. The leading case is testing hypotheses about the mean from a normal distribution (under random sampling) against a two-sided alternative. 47 7 . Computing p- values ∙ Classical hypothesis testing is somewhat cumbersome because it requires specifying a size ahead of time. Yet different researchers may have different tolerances for Type I errors. ∙ Moreover, in complicated settings, classical testing allows one to ignore a “close” rejection and just report that H was not rejected at, say, the 5% significance level. But it could have been rejected at the 6% level. 48 ∙ We can avoid choosing a size ahead of time by reporting a p- value with any test. This approach requires all the steps of classical testing except choosing a size. ∙ For illustration, consider testing H : 0 against H 1 : 0 in a Normal ,1 population under random sampling. Let T n X ̄ be the test statistic so that, under H , T Normal 0,1 . ∙ Now suppose we obtain the data and the value of the test statistic is t . If t ≤ 0 (which means x ̄ ≤ the test provides no evidence in favor of H 1 ; we fail to reject H in favor of H 1 at any reasonable size. 49 ∙ If t 0, for classical testing we would compare it with a relevant critical value one we have chosen the size of the test. Instead, consider the following calculation: P T t | P T t | H That is, if the null hypothesis is true, what is the probability of obvserving a statistic at least as large as the one we did observe? This is called the p-value of the test. Sometimes it is called a one- sided p- value . 50 ∙ The p-value nicely summarizes the evidence against the null hypothesis. It allows us to carry out a test at any significance level....
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∙ As we will see UMP tests generally do not exist against...

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